[seqfan] Some number triangles

Dale Gerdemann dale.gerdemann at gmail.com
Thu Apr 9 15:35:38 CEST 2015

```Dear SeqFans,

I've collected here some number triangles whose row sums are OEIS
sequences. They all follow a basic parameterized formula. Most of these
triangles are not in OEIS, and those that are in OEIS are usually defined
with rather different formulae.

There are also related triangles with interesting diagonal sums. For these,
I made a short video. There are 5 versions of the video, but they differ
only in coloring:

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 1*x + 1

Triangle: A008292

1

1 1

1 4 1

1 11 11 1

1 26 66 26 1

1 57 302 302 57 1

1 120 1191 2416 1191 120 1

1 247 4293 15619 15619 4293 247 1

1 502 14608 88234 156190 88234 14608 502 1

1 1013 47840 455192 1310354 1310354 455192 47840 1013 1

1 2036 152637 2203488 9738114 15724248 9738114 2203488 152637 2036 1

Row sums=A000142:[1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800,
39916800, 479001600, 6227020800]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 1

Triangle: A060187

1

1 1

1 6 1

1 23 23 1

1 76 230 76 1

1 237 1682 1682 237 1

1 722 10543 23548 10543 722 1

1 2179 60657 259723 259723 60657 2179 1

1 6552 331612 2485288 4675014 2485288 331612 6552 1

1 19673 1756340 21707972 69413294 69413294 21707972 1756340 19673 1

Row sums=A000165:[1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920,
185794560, 3715891200, 81749606400]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 1*x + 2

Triangle: not in OEIS

1

2 2

4 12 4

8 52 52 8

16 196 416 196 16

32 684 2644 2644 684 32

64 2276 14680 26440 14680 2276 64

128 7340 74652 220280 220280 74652 7340 128

256 23172 357328 1623964 2643360 1623964 357328 23172 256

512 72076 1637860 10978444 27227908 27227908 10978444 1637860 72076 512

Row sums=A001715:[1, 4, 20, 120, 840, 6720, 60480, 604800, 6652800,
79833600, 1037836800, 14529715200]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 1

Triangle: A142458

1

1 1

1 8 1

1 39 39 1

1 166 546 166 1

1 677 5482 5482 677 1

1 2724 47175 109640 47175 2724 1

1 10915 373809 1709675 1709675 373809 10915 1

1 43682 2824048 23077694 44451550 23077694 2824048 43682 1

1 174753 20729340 284433852 947113254 947113254 284433852 20729340 174753 1

Row sums=A008544:[1, 2, 10, 80, 880, 12320, 209440, 4188800, 96342400,
2504902400, 72642169600, 2324549427200]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 1*x + 3

Triangle: not in OEIS

1

3 3

9 24 9

27 141 141 27

81 726 1410 726 81

243 3471 11406 11406 3471 243

729 15828 81327 136872 81327 15828 729

2187 69873 533259 1390521 1390521 533259 69873 2187

6561 301362 3295152 12609198 19467294 12609198 3295152 301362 6561

Row sums=A001725:[1, 6, 42, 336, 3024, 30240, 332640, 3991680, 51891840,
726485760, 10897286400]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 1

Triangle: A142459

1

1 1

1 10 1

1 59 59 1

1 308 1062 308 1

1 1557 13562 13562 1557 1

1 7806 148527 352612 148527 7806 1

1 39055 1500669 7108915 7108915 1500669 39055 1

1 195304 14482396 123929944 241703110 123929944 14482396 195304 1

Row sums=A001813:[1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400,
17643225600, 670442572800]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 1*x + 4

Triangle: not in OEIS

1

4 4

16 40 16

64 296 296 64

256 1928 3552 1928 256

1024 11688 34808 34808 11688 1024

4096 67656 302352 487312 302352 67656 4096

16384 379240 2423016 5830000 5830000 2423016 379240 16384

65536 2076424 18330496 62617144 93280000 62617144 18330496 2076424 65536

Row sums=A049388:[1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200,
4151347200, 70572902400]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 5*x + 1

Triangle: A142460

1

1 1

1 12 1

1 83 83 1

1 514 1826 514 1

1 3105 28310 28310 3105 1

1 18656 376615 905920 376615 18656 1

1 111967 4627821 22403635 22403635 4627821 111967 1

1 671838 54377008 478781506 940952670 478781506 54377008 671838 1

Row sums=A047055:[1, 2, 14, 168, 2856, 62832, 1696464, 54286848,
2008613376, 84361761792, 3965002804224]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 1*x + 5

Triangle: not in OEIS

1

5 5

25 60 25

125 535 535 125

625 4210 7490 4210 625

3125 30885 86110 86110 30885 3125

15625 216560 880735 1377760 880735 216560 15625

78125 1471235 8330745 18948695 18948695 8330745 1471235 78125

390625 9764910 74498800 234897010 341076510 234897010 74498800 9764910
390625

Row sums=A049198:[1, 10, 110, 1320, 17160, 240240, 3603600, 57657600,
980179200, 17643225600, 335221286400]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 1

Triangle: A142461

1

1 1

1 14 1

1 111 111 1

1 796 2886 796 1

1 5597 52642 52642 5597 1

1 39210 824271 2000396 824271 39210 1

1 274507 11931033 58614299 58614299 11931033 274507 1

1 1921592 165260188 1483533704 2930714950 1483533704 165260188 1921592 1

Row sums=A047657:[1, 2, 16, 224, 4480, 116480, 3727360, 141639680,
6232145920, 311607296000, 17450008576000]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 1

Triangle: A142462

1

1 1

1 16 1

1 143 143 1

1 1166 4290 1166 1

1 9357 90002 90002 9357 1

1 74892 1621383 3960088 1621383 74892 1

1 599179 27016857 134142043 134142043 27016857 599179 1

1 4793482 431017552 3923731798 7780238494 3923731798 431017552 4793482 1

Row sums=A084947:[1, 2, 18, 288, 6624, 198720, 7352640, 323516160,
16499324160, 956960801280, 62202452083200]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 1

Triangle: A167884

1

1 1

1 18 1

1 179 179 1

1 1636 6086 1636 1

1 14757 144362 144362 14757 1

1 132854 2941135 7218100 2941135 132854 1

1 1195735 55446309 277509955 277509955 55446309 1195735 1

1 10761672 1001178268 9211047544 18315657030 9211047544 1001178268 10761672
1

Row sums=A084948:[1, 2, 20, 360, 9360, 318240, 13366080, 668304000,
38761632000, 2558267712000, 189311810688000]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1

Triangle: not in OEIS

1

1 1

1 20 1

1 219 219 1

1 2218 8322 2218 1

1 22217 220222 220222 22217 1

1 222216 5006247 12332432 5006247 222216 1

1 2222215 105340629 530539235 530539235 105340629 2222215 1

1 22222214 2123693776 19700767514 39259903390 19700767514 2123693776
22222214 1

Row sums=A084949:[1, 2, 22, 440, 12760, 484880, 22789360, 1276204160,
82953270400, 6138542009600, 509498986796800]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2

Triangle: not in OEIS

1

2 2

4 16 4

8 88 88 8

16 416 1056 416 16

32 1824 9664 9664 1824 32

64 7680 76224 154624 76224 7680 64

128 31616 549504 1999232 1999232 549504 31616 128

256 128512 3739648 22587904 39984640 22587904 3739648 128512 256

512 518656 24494080 233058304 670901248 670901248 233058304 24494080 518656
512

Row sums=A002866:[1, 4, 24, 192, 1920, 23040, 322560, 5160960, 92897280,
1857945600, 40874803200, 980995276800]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 2

Triangle: not in OEIS

1

2 2

4 20 4

8 132 132 8

16 748 2112 748 16

32 3964 25124 25124 3964 32

64 20364 256488 552728 256488 20364 64

128 103100 2398092 9670840 9670840 2398092 103100 128

256 518444 21246736 147146804 270783520 147146804 21246736 518444 256

Row sums=A007559:[1, 4, 28, 280, 3640, 58240, 1106560, 24344320, 608608000,
17041024000, 528271744000]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 3

Triangle: not in OEIS

1

3 3

9 30 9

27 213 213 27

81 1308 2982 1308 81

243 7431 32646 32646 7431 243

729 40314 310263 587628 310263 40314 729

2187 212505 2695923 8701545 8701545 2695923 212505 2187

6561 1099704 22059036 113360904 191433990 113360904 22059036 1099704 6561

Row sums=A051578:[1, 6, 48, 480, 5760, 80640, 1290240, 23224320, 464486400,
10218700800, 245248819200]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2

Triangle: not in OEIS

1

2 2

4 24 4

8 184 184 8

16 1216 3680 1216 16

32 7584 53824 53824 7584 32

64 46208 674752 1507072 674752 46208 64

128 278912 7764096 33244544 33244544 7764096 278912 128

256 1677312 84892672 636233728 1196803584 636233728 84892672 1677312 256

Row sums=A047053:[1, 4, 32, 384, 6144, 122880, 2949120, 82575360,
2642411520, 95126814720, 3805072588800]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4

Triangle: not in OEIS

1

4 4

16 48 16

64 416 416 64

256 3136 6656 3136 256

1024 21888 84608 84608 21888 1024

4096 145664 939520 1692160 939520 145664 4096

16384 939520 9555456 28195840 28195840 9555456 939520 16384

65536 5932032 91475968 415734784 676700160 415734784 91475968 5932032 65536

Row sums=A051580:[1, 8, 80, 960, 13440, 215040, 3870720, 77414400,
1703116800, 40874803200, 1062744883200]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 5*x + 2

Triangle: not in OEIS

1

2 2

4 28 4

8 244 244 8

16 1844 5856 1844 16

32 13260 101620 101620 13260 32

64 93684 1511160 3455080 1511160 93684 64

128 657836 20663388 91981880 91981880 20663388 657836 128

256 4609588 269011408 2121603436 4047202720 2121603436 269011408 4609588 256

Row sums=A008546:[1, 4, 36, 504, 9576, 229824, 6664896, 226606464,
8837652096, 388856692224, 19053977918976]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5

Triangle: not in OEIS

1

5 5

25 70 25

125 715 715 125

625 6380 12870 6380 625

3125 52785 186010 186010 52785 3125

15625 416370 2360295 4092220 2360295 416370 15625

78125 3180215 27488205 75698255 75698255 27488205 3180215 78125

Row sums=A051582:[1, 10, 120, 1680, 26880, 483840, 9676800, 212889600,
5109350400, 132843110400]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2

Triangle: not in OEIS

1

2 2

4 32 4

8 312 312 8

16 2656 8736 2656 16

32 21664 175424 175424 21664 32

64 174336 3019200 7016960 3019200 174336 64

128 1397120 47847552 218838400 218838400 47847552 1397120 128

256 11182592 722956288 5907889664 11379596800 5907889664 722956288 11182592
256

Row sums=A049308:[1, 4, 40, 640, 14080, 394240, 13404160, 536166400,
24663654400, 1282510028800, 74385581670400]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 2

Triangle: not in OEIS

1

2 2

4 36 4

8 388 388 8

16 3676 12416 3676 16

32 33564 283204 283204 33564 32

64 303260 5538184 13027384 5538184 303260 64

128 2732156 99831564 465775352 465775352 99831564 2732156 128

Row sums=A144827:[1, 4, 44, 792, 19800, 633600, 24710400, 1136678400,
60243955200, 3614637312000]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2

Triangle: not in OEIS

1

2 2

4 40 4

8 472 472 8

16 4928 16992 4928 16

32 49824 433984 433984 49824 32

64 499584 9505728 22567168 9505728 499584 64

128 4999040 192085632 909941120 909941120 192085632 4999040 128

Row sums=A144828:[1, 4, 48, 960, 26880, 967680, 42577920, 2214051840,
132843110400, 9033331507200]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 2

Triangle: not in OEIS

1

2 2

4 44 4

8 564 564 8

16 6436 22560 6436 16

32 71404 637844 637844 71404 32

64 786948 15470232 36994952 15470232 786948 64

128 8660012 346391196 1660722424 1660722424 346391196 8660012 128

Row sums=A144829:[1, 4, 52, 1144, 35464, 1418560, 69509440, 4031547520,
270113683840, 20528639971840]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 3

Triangle: not in OEIS

1

3 3

9 36 9

27 297 297 27

81 2106 5346 2106 81

243 13851 73386 73386 13851 243

729 87480 868239 1761264 868239 87480 729

2187 540189 9388791 34158753 34158753 9388791 540189 2187

6561 3293622 95843088 578903274 1024762590 578903274 95843088 3293622 6561

Row sums=A034001:[1, 6, 54, 648, 9720, 174960, 3674160, 88179840,
2380855680, 71425670400, 2357047123200]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 3

Triangle: not in OEIS

1

3 3

9 42 9

27 393 393 27

81 3156 8646 3156 81

243 23631 142446 142446 23631 243

729 171006 2015895 4273380 2015895 171006 729

2187 1216725 26107983 102402705 102402705 26107983 1216725 2187

6561 8584872 320039388 2136524184 3891302790 2136524184 320039388 8584872
6561

Row sums=A000407:[1, 6, 60, 840, 15120, 332640, 8648640, 259459200,
8821612800, 335221286400, 14079294028800]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4

Triangle: not in OEIS

1

4 4

16 56 16

64 552 552 64

256 4696 11040 4696 256

1024 36968 171448 171448 36968 1024

4096 278232 2305968 4457648 2305968 278232 4096

16384 2037736 28346088 94844912 94844912 28346088 2037736 16384

Row sums=A051605:[1, 8, 88, 1232, 20944, 418880, 9634240, 250490240,
7264216960, 232454942720]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 5*x + 3

Triangle: not in OEIS

1

3 3

9 48 9

27 501 501 27

81 4494 13026 4494 81

243 37815 250230 250230 37815 243

729 309324 4122735 9008280 4122735 309324 729

2187 2498649 62256627 256971945 256971945 62256627 2498649 2187

Row sums=A008548:[1, 6, 66, 1056, 22176, 576576, 17873856, 643458816,
26381811456, 1213563326976]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 5

Triangle: not in OEIS

1

5 5

25 80 25

125 915 915 125

625 9070 20130 9070 625

3125 83185 348410 348410 83185 3125

15625 727980 5246655 9755480 5246655 727980 15625

78125 6183215 72272805 225769855 225769855 72272805 6183215 78125

Row sums=A051607:[1, 10, 130, 2080, 39520, 869440, 21736000, 608608000,
18866848000, 641472832000]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 3

Triangle: not in OEIS

1

3 3

9 54 9

27 621 621 27

81 6156 18630 6156 81

243 57591 408726 408726 57591 243

729 526338 7685847 17166492 7685847 526338 729

2187 4765473 132656859 568014201 568014201 132656859 4765473 2187

Row sums=A047058:[1, 6, 72, 1296, 31104, 933120, 33592320, 1410877440,
67722117120, 3656994324480]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6

Triangle: not in OEIS

1

6 6

36 108 36

216 1404 1404 216

1296 15876 33696 15876 1296

7776 166212 642492 642492 166212 7776

46656 1659204 10701720 19274760 10701720 1659204 46656

279936 16052580 163263924 481752360 481752360 163263924 16052580 279936

Row sums=A051609:[1, 12, 180, 3240, 68040, 1632960, 44089920, 1322697600,
43649020800, 1571364748800]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 3

Triangle: not in OEIS

1

3 3

9 60 9

27 753 753 27

81 8178 25602 8178 81

243 84291 631506 631506 84291 243

729 852144 13348623 30312288 13348623 852144 729

2187 8554245 259308063 1141302225 1141302225 259308063 8554245 2187

Row sums=A049209:[1, 6, 78, 1560, 42120, 1432080, 58715280, 2818333440,
155008339200, 9610517030400]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 3

Triangle: not in OEIS

1

3 3

9 66 9

27 897 897 27

81 10596 34086 10596 81

243 119391 933726 933726 119391 243

729 1323750 21919479 50421204 21919479 1323750 729

2187 14598429 473391351 2128554273 2128554273 473391351 14598429 2187

Row sums=A147626:[1, 6, 84, 1848, 55440, 2106720, 96909120, 5233092480,
324451733760, 22711621363200]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 3

Triangle: not in OEIS

1

3 3

9 72 9

27 1053 1053 27

81 13446 44226 13446 81

243 164511 1332126 1332126 164511 243

729 1985796 34390575 79927560 34390575 1985796 729

2187 23871105 817520283 3739059225 3739059225 817520283 23871105 2187

Row sums=A147630:[1, 6, 90, 2160, 71280, 2993760, 152681760, 9160905600,
632102486400, 49303993939200]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 4

Triangle: not in OEIS

1

4 4

16 64 16

64 704 704 64

256 6656 16896 6656 256

1024 58368 309248 309248 58368 1024

4096 491520 4878336 9895936 4878336 491520 4096

16384 4046848 70336512 255901696 255901696 70336512 4046848 16384

Row sums=A034177:[1, 8, 96, 1536, 30720, 737280, 20643840, 660602880,
23781703680, 951268147200]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 5*x + 4

Triangle: not in OEIS

1

4 4

16 72 16

64 872 872 64

256 9064 24416 9064 256

1024 87720 514040 514040 87720 1024

4096 819176 9301840 19533520 9301840 819176 4096

16384 7511848 153981864 594381040 594381040 153981864 7511848 16384

Row sums=A034300:[1, 8, 104, 1872, 43056, 1205568, 39783744, 1511782272,
65006637696, 3120318609408]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 5

Triangle: not in OEIS

1

5 5

25 90 25

125 1135 1135 125

625 12340 29510 12340 625

3125 124185 593410 593410 124185 3125

15625 1195790 10322215 20175940 10322215 1195790 15625

78125 11215235 164083545 559757495 559757495 164083545 11215235 78125

Row sums=A051618:[1, 10, 140, 2520, 55440, 1441440, 43243200, 1470268800,
55870214400, 2346549004800]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 4

Triangle: not in OEIS

1

4 4

16 80 16

64 1056 1056 64

256 11968 33792 11968 256

1024 126848 803968 803968 126848 1024

4096 1303296 16415232 35374592 16415232 1303296 4096

16384 13196800 306955776 1237867520 1237867520 306955776 13196800 16384

Row sums=A034689:[1, 8, 112, 2240, 58240, 1863680, 70819840, 3116072960,
155803648000, 8725004288000]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 6

Triangle: not in OEIS

1

6 6

36 120 36

216 1704 1704 216

1296 20928 47712 20928 1296

7776 237792 1044672 1044672 237792 7776

46656 2580096 19856832 37608192 19856832 2580096 46656

279936 27200640 345078144 1113797760 1113797760 345078144 27200640 279936

Row sums=A051620:[1, 12, 192, 3840, 92160, 2580480, 82575360, 2972712960,
118908518400, 5231974809600]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 4

Triangle: not in OEIS

1

4 4

16 88 16

64 1256 1256 64

256 15416 45216 15416 256

1024 177768 1199288 1199288 177768 1024

4096 1995384 27275760 59964400 27275760 1995384 4096

16384 22137640 568783656 2371934320 2371934320 568783656 22137640 16384

Row sums=A045754:[1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000,
337767408000, 21617114112000]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 7

Triangle: not in OEIS

1

7 7

49 154 49

343 2429 2429 343

2401 33236 72870 33236 2401

16807 420819 1724534 1724534 420819 16807

117649 5082798 35546847 65532292 35546847 5082798 117649

823543 59557897 670438251 2062691029 2062691029 670438251 59557897 823543

Row sums=A051622:[1, 14, 252, 5544, 144144, 4324320, 147026880, 5587021440,
234654900480, 10794125422080]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 4

Triangle: not in OEIS

1

4 4

16 96 16

64 1472 1472 64

256 19456 58880 19456 256

1024 242688 1722368 1722368 242688 1024

4096 2957312 43184128 96452608 43184128 2957312 4096

16384 35700736 993804288 4255301632 4255301632 993804288 35700736 16384

Row sums=A051189:[1, 8, 128, 3072, 98304, 3932160, 188743680, 10569646080,
676457349120, 48704929136640]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 4

Triangle: not in OEIS

1

4 4

16 104 16

64 1704 1704 64

256 24136 74976 24136 256

1024 324008 2397688 2397688 324008 1024

4096 4262280 65709456 148656656 65709456 4262280 4096

16384 55647208 1654459752 7236734576 7236734576 1654459752 55647208 16384

Row sums=A049211:[1, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840,
1270453824640, 101636305971200]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 5*x + 5

Triangle: not in OEIS

1

5 5

25 100 25

125 1375 1375 125

625 16250 41250 16250 625

3125 178125 943750 943750 178125 3125

15625 1875000 18609375 37750000 18609375 1875000 15625

78125 19296875 335390625 1220234375 1220234375 335390625 19296875 78125

Row sums=A034325:[1, 10, 150, 3000, 75000, 2250000, 78750000, 3150000000,
141750000000, 7087500000000]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 5

Triangle: not in OEIS

1

5 5

25 110 25

125 1635 1635 125

625 20860 55590 20860 625

3125 247585 1424810 1424810 247585 3125

15625 2832810 31401735 65541260 31401735 2832810 15625

78125 31801535 632977845 2418099295 2418099295 632977845 31801535 78125

Row sums=A034724:[1, 10, 160, 3520, 98560, 3351040, 134041600, 6165913600,
320627507200, 18596395417600]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 5*x + 6

Triangle: not in OEIS

1

6 6

36 132 36

216 2028 2028 216

1296 26844 64896 26844 1296

7776 328980 1602060 1602060 328980 7776

46656 3859836 34186440 67286520 34186440 3859836 46656

279936 44137812 666637956 2301864360 2301864360 666637956 44137812 279936

Row sums=A051688:[1, 12, 204, 4488, 121176, 3877632, 143472384, 6025840128,
283214486016, 14727153272832]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 5

Triangle: not in OEIS

1

5 5

25 120 25

125 1915 1915 125

625 26230 72770 26230 625

3125 335385 2064610 2064610 335385 3125

15625 4149620 50295295 107359720 50295295 4149620 15625

78125 50529815 1121595405 4451097455 4451097455 1121595405 50529815 78125

Row sums=A034830:[1, 10, 170, 4080, 126480, 4806240, 216280800,
11246601600, 663549494400, 43794266630400]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 5*x + 7

Triangle: not in OEIS

1

7 7

49 168 49

343 2849 2849 343

2401 41734 96866 41734 2401

16807 565635 2564870 2564870 565635 16807

117649 7325444 58874935 112854280 58874935 7325444 117649

823543 92258341 1235288103 4072417405 4072417405 1235288103 92258341 823543

Row sums=A051690:[1, 14, 266, 6384, 185136, 6294624, 245490336,
10801574784, 529277164416, 28580966878464]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 5

Triangle: not in OEIS

1

5 5

25 130 25

125 2215 2215 125

625 32420 93030 32420 625

3125 444585 2893810 2893810 444585 3125

15625 5920230 77219655 167840980 77219655 5920230 15625

78125 77791115 1888023105 7724515655 7724515655 1888023105 77791115 78125

Row sums=A034908:[1, 10, 180, 4680, 159120, 6683040, 334152000,
19380816000, 1279133856000, 94655905344000]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 5

Triangle: not in OEIS

1

5 5

25 140 25

125 2535 2535 125

625 39490 116610 39490 625

3125 578485 3945710 3945710 578485 3125

15625 8255040 114469215 252525440 114469215 8255040 15625

78125 116492435 3045543945 12774051895 12774051895 3045543945 116492435
78125

Row sums=A045756:[1, 10, 190, 5320, 196840, 9054640, 498005200,
31872332800, 2326680294400, 190787784140800]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 10*x + 5

Triangle: not in OEIS

1

5 5

25 150 25

125 2875 2875 125

625 47500 143750 47500 625

3125 740625 5256250 5256250 740625 3125

15625 11281250 164734375 367937500 164734375 11281250 15625

78125 170234375 4738828125 20290859375 20290859375 4738828125 170234375
78125

Row sums=A051262:[1, 10, 200, 6000, 240000, 12000000, 720000000,
50400000000, 4032000000000, 362880000000000]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 11*x + 5

Triangle: not in OEIS

1

5 5

25 160 25

125 3235 3235 125

625 56510 174690 56510 625

3125 934785 6864010 6864010 934785 3125

15625 15144060 231132735 521664760 231132735 15144060 15625

78125 243414335 7149227445 31148764895 31148764895 7149227445 243414335
78125

Row sums=A254322:[1, 10, 210, 6720, 288960, 15603840, 1014249600,
77082969600, 6706218355200, 657209398809600]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 6

Triangle: not in OEIS

1

6 6

36 144 36

216 2376 2376 216

1296 33696 85536 33696 1296

7776 443232 2348352 2348352 443232 7776

46656 5598720 55567296 112720896 55567296 5598720 46656

279936 69144192 1201765248 4372320384 4372320384 1201765248 69144192 279936

Row sums=A034788:[1, 12, 216, 5184, 155520, 5598720, 235146240,
11287019520, 609499054080, 36569943244800]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 6

Triangle: not in OEIS

1

6 6

36 156 36

216 2748 2748 216

1296 41556 109920 41556 1296

7776 584292 3320412 3320412 584292 7776

46656 7914612 86274168 179302248 86274168 7914612 46656

279936 105129444 2049982452 7774482408 7774482408 2049982452 105129444
279936

Row sums=A034832:[1, 12, 228, 5928, 195624, 7824960, 367773120,
19859748480, 1211444657280, 82378236695040]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 7

Triangle: not in OEIS

1

7 7

49 182 49

343 3297 3297 343

2401 51436 125286 51436 2401

16807 743099 3666334 3666334 743099 16807

117649 10282146 92696415 183316700 92696415 10282146 117649

823543 138726805 2141671287 7456506365 7456506365 2141671287 138726805
823543

Row sums=A053101:[1, 14, 280, 7280, 232960, 8852480, 389509120,
19475456000, 1090625536000, 67618783232000]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 6

Triangle: not in OEIS

1

6 6

36 168 36

216 3144 3144 216

1296 50496 138336 50496 1296

7776 756192 4558272 4558272 756192 7776

46656 10944384 129017280 273496320 129017280 10944384 46656

279936 155740800 3341821824 13107546240 13107546240 3341821824 155740800
279936

Row sums=A034910:[1, 12, 240, 6720, 241920, 10644480, 553512960,
33210777600, 2258332876800, 171633298636800]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 8

Triangle: not in OEIS

1

8 8

64 224 64

512 4416 4416 512

4096 75136 176640 75136 4096

32768 1182976 5486336 5486336 1182976 32768

262144 17806848 147581952 285289472 147581952 17806848 262144

Row sums=A053103:[1, 16, 352, 9856, 335104, 13404160, 616591360,
32062750720, 1859639541760]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 6

Triangle: not in OEIS

1

6 6

36 180 36

216 3564 3564 216

1296 60588 171072 60588 1296

7776 963252 6105132 6105132 963252 7776

46656 14845356 186979752 402938712 186979752 14845356 46656

279936 225479700 5244627204 21150127080 21150127080 5244627204 225479700
279936

Row sums=A035013:[1, 12, 252, 7560, 294840, 14152320, 806682240,
53241027840, 3993077088000, 335418475392000]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 10*x + 6

Triangle: not in OEIS

1

6 6

36 192 36

216 4008 4008 216

1296 71904 208416 71904 1296

7776 1210080 8007360 8007360 1210080 7776

46656 19796736 263855040 576529920 263855040 19796736 46656

279936 319827072 7968848256 32892408960 32892408960 7968848256 319827072
279936

Row sums=A035265:[1, 12, 264, 8448, 354816, 18450432, 1143926784,
82362728448, 6753743732736, 621344423411712]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 12*x + 6

Triangle: not in OEIS

1

6 6

36 216 36

216 4968 4968 216

1296 98496 298080 98496 1296

7776 1842912 13079232 13079232 1842912 7776

46656 33685632 491894208 1098655488 491894208 33685632 46656

Row sums=A145448:[1, 12, 288, 10368, 497664, 29859840, 2149908480,
180592312320, 17336861982720]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 7

Triangle: not in OEIS

1

7 7

49 196 49

343 3773 3773 343

2401 62426 158466 62426 2401

16807 957999 5075714 5075714 957999 16807

117649 14117880 140119959 284239984 140119959 14117880 117649

823543 203415121 3535470099 12862918117 12862918117 3535470099 203415121
823543

Row sums=A034834:[1, 14, 294, 8232, 288120, 12101040, 592950960,
33205253760, 2091930986880, 146435169081600]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 7

Triangle: not in OEIS

1

7 7

49 210 49

343 4277 4277 343

2401 74788 196742 74788 2401

16807 1215459 6843494 6843494 1215459 16807

117649 19021814 204803263 424296628 204803263 19021814 117649

823543 291797905 5604500307 21140522725 21140522725 5604500307 291797905
823543

Row sums=A034912:[1, 14, 308, 9240, 351120, 16151520, 872182080,
54075288960, 3785270227200, 295251077721600]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 8

Triangle: not in OEIS

1

8 8

64 240 64

512 5008 5008 512

4096 89968 220352 89968 4096

32768 1496976 7456816 7456816 1496976 32768

262144 23863664 217941088 432495328 217941088 23863664 262144

Row sums=A053105:[1, 16, 368, 11040, 408480, 17973120, 916629120,
53164488960, 3455691782400]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 7

Triangle: not in OEIS

1

7 7

49 224 49

343 4809 4809 343

2401 88606 240450 88606 2401

16807 1520939 9023854 9023854 1520939 16807

117649 25208988 290996727 613622072 290996727 25208988 117649

823543 410520397 8585785551 33376009709 33376009709 8585785551 410520397
823543

Row sums=A035018:[1, 14, 322, 10304, 422464, 21123200, 1246268800,
84746278400, 6525463436800, 561189855564800]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 10*x + 7

Triangle: not in OEIS

1

7 7

49 238 49

343 5369 5369 343

2401 103964 289926 103964 2401

16807 1880235 11674670 11674670 1880235 16807

117649 32921994 403587135 863925580 403587135 32921994 117649

Row sums=A035273:[1, 14, 336, 11424, 502656, 27143424, 1737179136,
128551256064, 10798305509376]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 8

Triangle: not in OEIS

1

8 8

64 256 64

512 5632 5632 512

4096 106496 270336 106496 4096

32768 1867776 9895936 9895936 1867776 32768

262144 31457280 312213504 633339904 312213504 31457280 262144

Row sums=A034976:[1, 16, 384, 12288, 491520, 23592960, 1321205760,
84557168640, 6088116142080]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 8

Triangle: not in OEIS

1

8 8

64 272 64

512 6288 6288 512

4096 124816 326976 124816 4096

32768 2302096 12869936 12869936 2302096 32768

262144 40872336 435910560 900895520 435910560 40872336 262144

Row sums=A035021:[1, 16, 400, 13600, 584800, 30409600, 1854985600,
129848992000, 10258070368000]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 9

Triangle: not in OEIS

1

9 9

81 306 81

729 7227 7227 729

6561 146916 361350 146916 6561

59049 2766573 13881978 13881978 2766573 59049

531441 49925142 460478943 916210548 460478943 49925142 531441

Row sums=A053115:[1, 18, 468, 15912, 668304, 33415200, 1938081600,
127913385600, 9465590534400]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 9

Triangle: not in OEIS

1

9 9

81 324 81

729 8019 8019 729

6561 170586 433026 170586 6561

59049 3365793 17832798 17832798 3365793 59049

531441 63772920 632946231 1283961456 632946231 63772920 531441

Row sums=A035023:[1, 18, 486, 17496, 787320, 42515280, 2678462640,
192849310080, 15620794116480]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 10*x + 9

Triangle: not in OEIS

1

9 9

81 342 81

729 8847 8847 729

6561 196524 513126 196524 6561

59049 4055445 22545090 22545090 4055445 59049

531441 80537346 852524415 1758517020 852524415 80537346 531441

Row sums=A035277:[1, 18, 504, 19152, 919296, 53319168, 3625703424,
282804867072, 24886828302336]

Formula: T(n,k) = t(n-k,k), t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0,

t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 10*x + 10

Triangle: not in OEIS

1

10 10

100 400 100

1000 11000 11000 1000

10000 260000 660000 260000 10000

100000 5700000 30200000 30200000 5700000 100000

1000000 120000000 1191000000 2416000000 1191000000 120000000 1000000

Row sums=A035279:[1, 20, 600, 24000, 1200000, 72000000, 5040000000,
403200000000, 36288000000000]

```