lost submissions to OEIS

Sat Apr 26 16:05:17 CEST 2008

On Sat, Apr 26, 2008 at 3:29 AM, Max Alekseyev <maxale at gmail.com> wrote:
>  A particular one that has drawn my attention is about "factorable trinomials":
>  If anybody wants to elaborate on this topic and add new
>  references/sequences to OEIS - please do.

Concerning the trinomials, as said in these Notes, the coefficient of
x in the series for which P[1]=1-ax+bx^2
is given by the expansion of 1/(1-ax+bx^2), viz

gp> FT(a,b=1,n=30)=Vec(1/(1-a*x+b*x^2+O(x^n)))
e.g.
gp> FT(-2,2)
%308 = [1, -2, 2, 0, -4, 8, -8, 0, 16, -32, 32, 0, -64, 128, -128, 0, ....]

As it is the case for this example (A108520),
I suspect many of these sequences are already in OEIS,
but it would be interesting to add a related comment to all of the
concerned sequences,
or better, a link to the list of related sequences which could be
compiled in a page of the index ;
either under "Trinomials", or under "Factorable"
(Neil: there's a typo in the 2nd last line of
http://www.research.att.com/~njas/sequences/Sindx_Fa.html)
then the reference and comments as those given in these "Notes"
(recursion relation,...)
could be kept grouped together in one place (avoiding content
duplication and facilitating maintainance)

e.g. b=1:
a=0: A056594=A101455=[1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0,...
a=1: A010892=A128834(1,...)=[1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, ...
a=2: A000027=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...
a=3: A001906(1,...)=A088305(1,...)=[1, 3, 8, 21, 55, 144, 377, 987,
2584, 6765,..
a=4: A001353=[1, 4, 15, 56, 209, 780, 2911, 10864, 40545,...
a=5: A004254=[1, 5, 24, 115, 551, 2640, 12649, 60605, ...
a=6: A001109=[1, 6, 35, 204, 1189, 6930, 40391, 235416, ...
a=7: A004187=[1, 7, 48, 329, 2255, 15456, 105937, 726103,...
a=8: A001090=[1, 8, 63, 496, 3905, 30744, 242047, 1905632,...
a=9: A018913=[1, 9, 80, 711, 6319, 56160, 499121, 4435929,...
a=10: A004189=[1, 10, 99, 980, 9701, 96030, 950599, 9409960,...
...
actually, they might also be listed under "recurrent sequences".
Someone wants to continue collecting references?
For convenience, I put the values below.
Maximilian

negative "a": (only changes sign of every other term)
a=-1: [1, -1, 0, 1, -1, 0, 1, -1, 0, 1]
a=-2: [1, -2, 3, -4, 5, -6, 7, -8, 9, -10]
...[1, -3, 8, -21, 55, -144, 377, -987, 2584, -6765]
[1, -4, 15, -56, 209, -780, 2911, -10864, 40545, -151316]
[1, -5, 24, -115, 551, -2640, 12649, -60605, 290376, -1391275]
[1, -6, 35, -204, 1189, -6930, 40391, -235416, 1372105, -7997214]
[1, -7, 48, -329, 2255, -15456, 105937, -726103, 4976784, -34111385]
[1, -8, 63, -496, 3905, -30744, 242047, -1905632, 15003009, -118118440]
[1, -9, 80, -711, 6319, -56160, 499121, -4435929, 39424240, -350382231]
[1, -10, 99, -980, 9701, -96030, 950599, -9409960, 93149001, -922080050]

***** b=-1 ****
a=-5: [1, -5, 26, -135, 701, -3640, 18901, -98145, 509626, -2646275]
a=-4: [1, -4, 17, -72, 305, -1292, 5473, -23184, 98209, -416020]
a=-3: [1, -3, 10, -33, 109, -360, 1189, -3927, 12970, -42837]
a=-2: [1, -2, 5, -12, 29, -70, 169, -408, 985, -2378]
a=-1: [1, -1, 2, -3, 5, -8, 13, -21, 34, -55]
a=0: [1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
a=1: [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
a=2: [1, 2, 5, 12, 29, 70, 169, 408, 985, 2378]
a=3: [1, 3, 10, 33, 109, 360, 1189, 3927, 12970, 42837]
a=4: [1, 4, 17, 72, 305, 1292, 5473, 23184, 98209, 416020]
a=5: [1, 5, 26, 135, 701, 3640, 18901, 98145, 509626, 2646275]

**** b=2 ****
a=-5: [1, -5, 23, -105, 479, -2185, 9967, -45465, 207391, -946025]
a=-4: [1, -4, 14, -48, 164, -560, 1912, -6528, 22288, -76096]
a=-3: [1, -3, 7, -15, 31, -63, 127, -255, 511, -1023]
a=-2: [1, -2, 2, 0, -4, 8, -8, 0, 16, -32]
a=-1: [1, -1, -1, 3, -1, -5, 7, 3, -17, 11]
a=0: [1, 0, -2, 0, 4, 0, -8, 0, 16, 0]
a=1: [1, 1, -1, -3, -1, 5, 7, -3, -17, -11]
a=2: [1, 2, 2, 0, -4, -8, -8, 0, 16, 32]
a=3: [1, 3, 7, 15, 31, 63, 127, 255, 511, 1023]
a=4: [1, 4, 14, 48, 164, 560, 1912, 6528, 22288, 76096]
a=5: [1, 5, 23, 105, 479, 2185, 9967, 45465, 207391, 946025]
**** b=3 ****
a=-5: [1, -5, 22, -95, 409, -1760, 7573, -32585, 140206, -603275]
a=-4: [1, -4, 13, -40, 121, -364, 1093, -3280, 9841, -29524]
a=-3: [1, -3, 6, -9, 9, 0, -27, 81, -162, 243]
a=-2: [1, -2, 1, 4, -11, 10, 13, -56, 73, 22]
a=-1: [1, -1, -2, 5, 1, -16, 13, 35, -74, -31]
a=0: [1, 0, -3, 0, 9, 0, -27, 0, 81, 0]
a=1: [1, 1, -2, -5, 1, 16, 13, -35, -74, 31]
a=2: [1, 2, 1, -4, -11, -10, 13, 56, 73, -22]
a=3: [1, 3, 6, 9, 9, 0, -27, -81, -162, -243]
a=4: [1, 4, 13, 40, 121, 364, 1093, 3280, 9841, 29524]
a=5: [1, 5, 22, 95, 409, 1760, 7573, 32585, 140206, 603275]
**** b=4 ****
a=-5: [1, -5, 21, -85, 341, -1365, 5461, -21845, 87381, -349525]
a=-4: [1, -4, 12, -32, 80, -192, 448, -1024, 2304, -5120]
a=-3: [1, -3, 5, -3, -11, 45, -91, 93, 85, -627]
a=-2: [1, -2, 0, 8, -16, 0, 64, -128, 0, 512]
a=-1: [1, -1, -3, 7, 5, -33, 13, 119, -171, -305]
a=0: [1, 0, -4, 0, 16, 0, -64, 0, 256, 0]
a=1: [1, 1, -3, -7, 5, 33, 13, -119, -171, 305]
a=2: [1, 2, 0, -8, -16, 0, 64, 128, 0, -512]
a=3: [1, 3, 5, 3, -11, -45, -91, -93, 85, 627]
a=4: [1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120]
a=5: [1, 5, 21, 85, 341, 1365, 5461, 21845, 87381, 349525]
**** b=5 ****
a=-5: [1, -5, 20, -75, 275, -1000, 3625, -13125, 47500, -171875]
a=-4: [1, -4, 11, -24, 41, -44, -29, 336, -1199, 3116]
a=-3: [1, -3, 4, 3, -29, 72, -71, -147, 796, -1653]
a=-2: [1, -2, -1, 12, -19, -22, 139, -168, -359, 1558]
a=-1: [1, -1, -4, 9, 11, -56, 1, 279, -284, -1111]
a=0: [1, 0, -5, 0, 25, 0, -125, 0, 625, 0]
a=1: [1, 1, -4, -9, 11, 56, 1, -279, -284, 1111]
a=2: [1, 2, -1, -12, -19, 22, 139, 168, -359, -1558]
a=3: [1, 3, 4, -3, -29, -72, -71, 147, 796, 1653]
a=4: [1, 4, 11, 24, 41, 44, -29, -336, -1199, -3116]
a=5: [1, 5, 20, 75, 275, 1000, 3625, 13125, 47500, 171875]

**** b=-5 ****
a=-5: [1, -5, 30, -175, 1025, -6000, 35125, -205625, 1203750, -7046875]
a=-4: [1, -4, 21, -104, 521, -2604, 13021, -65104, 325521, -1627604]
a=-3: [1, -3, 14, -57, 241, -1008, 4229, -17727, 74326, -311613]
a=-2: [1, -2, 9, -28, 101, -342, 1189, -4088, 14121, -48682]
a=-1: [1, -1, 6, -11, 41, -96, 301, -781, 2286, -6191]
a=0: [1, 0, 5, 0, 25, 0, 125, 0, 625, 0]
a=1: [1, 1, 6, 11, 41, 96, 301, 781, 2286, 6191]
a=2: [1, 2, 9, 28, 101, 342, 1189, 4088, 14121, 48682]
a=3: [1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613]
a=4: [1, 4, 21, 104, 521, 2604, 13021, 65104, 325521, 1627604]
a=5: [1, 5, 30, 175, 1025, 6000, 35125, 205625, 1203750, 7046875]
**** b=-4 ****
a=-5: [1, -5, 29, -165, 941, -5365, 30589, -174405, 994381, -5669525]
a=-4: [1, -4, 20, -96, 464, -2240, 10816, -52224, 252160, -1217536]
a=-3: [1, -3, 13, -51, 205, -819, 3277, -13107, 52429, -209715]
a=-2: [1, -2, 8, -24, 80, -256, 832, -2688, 8704, -28160]
a=-1: [1, -1, 5, -9, 29, -65, 181, -441, 1165, -2929]
a=0: [1, 0, 4, 0, 16, 0, 64, 0, 256, 0]
a=1: [1, 1, 5, 9, 29, 65, 181, 441, 1165, 2929]
a=2: [1, 2, 8, 24, 80, 256, 832, 2688, 8704, 28160]
a=3: [1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715]
a=4: [1, 4, 20, 96, 464, 2240, 10816, 52224, 252160, 1217536]
a=5: [1, 5, 29, 165, 941, 5365, 30589, 174405, 994381, 5669525]
**** b=-3 ****
a=-5: [1, -5, 28, -155, 859, -4760, 26377, -146165, 809956, -4488275]
a=-4: [1, -4, 19, -88, 409, -1900, 8827, -41008, 190513, -885076]
a=-3: [1, -3, 12, -45, 171, -648, 2457, -9315, 35316, -133893]
a=-2: [1, -2, 7, -20, 61, -182, 547, -1640, 4921, -14762]
a=-1: [1, -1, 4, -7, 19, -40, 97, -217, 508, -1159]
a=0: [1, 0, 3, 0, 9, 0, 27, 0, 81, 0]
a=1: [1, 1, 4, 7, 19, 40, 97, 217, 508, 1159]
a=2: [1, 2, 7, 20, 61, 182, 547, 1640, 4921, 14762]
a=3: [1, 3, 12, 45, 171, 648, 2457, 9315, 35316, 133893]
a=4: [1, 4, 19, 88, 409, 1900, 8827, 41008, 190513, 885076]
a=5: [1, 5, 28, 155, 859, 4760, 26377, 146165, 809956, 4488275]
**** b=-2 ****
a=-5: [1, -5, 27, -145, 779, -4185, 22483, -120785, 648891, -3486025]
a=-4: [1, -4, 18, -80, 356, -1584, 7048, -31360, 139536, -620864]
a=-3: [1, -3, 11, -39, 139, -495, 1763, -6279, 22363, -79647]
a=-2: [1, -2, 6, -16, 44, -120, 328, -896, 2448, -6688]
a=-1: [1, -1, 3, -5, 11, -21, 43, -85, 171, -341]
a=0: [1, 0, 2, 0, 4, 0, 8, 0, 16, 0]
a=1: [1, 1, 3, 5, 11, 21, 43, 85, 171, 341]
a=2: [1, 2, 6, 16, 44, 120, 328, 896, 2448, 6688]
a=3: [1, 3, 11, 39, 139, 495, 1763, 6279, 22363, 79647]
a=4: [1, 4, 18, 80, 356, 1584, 7048, 31360, 139536, 620864]
a=5: [1, 5, 27, 145, 779, 4185, 22483, 120785, 648891, 3486025]