# successive differences of A001147 Double factorial numbers

Jonathan Post jvospost3 at gmail.com
Thu Jul 10 18:55:28 CEST 2008

```Is the following worthy of OEIS submission?  (Or the not shown
Triangular array formed from successive differences of A006882 Double
factorials n!!: a(n)=n*a(n-2))?

Analogue of A047920 Triangular array formed from successive
differences of factorial numbers.

Triangular array formed from successive differences of A001147 Double
factorial numbers: (2n-1)!! = 1.3.5....(2n-1)..

0: 1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425
1: 0, 2, 12, 90, 840, 9450, 124740, 1891890, 32432400
2: 2, 10, 78, 750, 8610, 115290, 1767150, 30540510
3: 8, 68, 7860, 106680, 1651860, 28773360
4: 60, 7792, 98820, 1545180, 27121500
5: 7732, 91028, 1446360, 25576320
6: 83296, 1355332, 24129960
7: 1272036, 22774628
8: 21502592

=============

First differences:

34459425 - 2027025 :
32432400 = 2^4 × 3^4 × 5^2 × 7 × 11 × 13

2027025 - 135135 :
1891890 = 2 × 3^3 × 5 × 7^2 × 11 × 13

135135 - 10395 :
124740 = 2^2 × 3^4 × 5 × 7 × 11

10395 - 945 :
9450 = 2 × 3^3 × 52 × 7

945 - 105 :
840 = 2^3 × 3 × 5 × 7

105 - 15 :
90 = 2 × 3^2 × 5

=============
Second differences:

32432400 - 1891890 :
30540510 = 2 × 3^3 × 5 × 7 × 11 × 13 × 113

1891890 - 124740 :
1767150 = 2 × 3^3 × 5^2 × 7 × 11 × 17

124740 - 9450 :
115290 = 2 × 3^3 × 5 × 7 × 61

9450 - 840 :
8610 = 2 × 3 × 5 × 7 × 41

840 - 90 :
750 = 2 × 3 × 5^3

90 - 12 :
78 = 2 × 3 × 13

=============
Third differences:

30540510 - 1767150 :
28773360 = 2^4 × 3^3 × 5 × 7 × 11 × 173

1767150 - 115290 :
1651860 = 2^2 × 3^3 × 5 × 7 × 19 × 23

115290 - 8610 :
106680 = 2^3 × 3 × 5 × 7 × 127

8610 - 750 :
7860 = 2^2 × 3 × 5 × 131

750 - 78 :
672 = 25 × 3 × 7

=============
Fourth differences:

28773360 - 1651860 :
27121500 = 2^2 × 3^3 × 5^3 × 7^2 × 41

1651860 - 106680 :
1545180 = 2^2 × 3 × 5 × 7 × 13 × 283

106680 - 7860 :
98820 = 2^2 × 3^4 × 5 × 61

7860 - 68 :
7792 = 2^4 × 487

==========
Fifth differences:

27121500 - 1545180 :
25576320 = 2^7 × 3 × 5 × 7 × 11 × 173

1545180 - 98820 :
1446360 = 2^3 × 3 × 5 × 17 × 709

98820 - 7792 :
91028 = 2^2 × 7 × 3251

7792 - 60 :
7732 = 2^2 × 1933

==========
Sixth differences:

25576320 - 1446360 :
24129960 = 2^3 × 3 × 5 × 211 × 953

1446360 - 91028 :
1355332 = 2^2 × 11 × 30803

91028 - 7732 :
83296 = 2^5 × 19 × 137

==========
Seventh differences:

24129960 - 1355332 :
22774628 = 2^2 × 17 × 29 × 11549

1355332 - 83296 :
1272036 = 2^2 × 3 × 71 × 1493
==========

Eighth differences:

22774628 - 1272036 :
21502592 = 2^7 × 31 × 5419

==========

X-SIG5: 69bab78a974f843dfd59bc7e159ddb15

<p>I think that the morpionsolitaire's sequences are eligible.</p><p><br />The game appears to be fascinating and it is possible that the game intrinsic winning algorithm have some hidden combinatorial applications. Graph drawing? Who knows?</p>

X-SIG5: 882a10361ae272af844b739cb67fb039

<p>The number of bracelets can be given by a sum involving the Euler totient<br />function, but in some cases there are very simple expressions that count those<br />objects.Â <br /><br />For example, I found in njas's sequence A001399 that the number of bracelets<br />with n+3 beads 3 of which are red is equal to the nearest integer to (n+3)^2/12.</p><p><br />There areÂ similar expressions for the case of 4 "reds", or "5 reds"?<br /><br />There isÂ a method to find simple expressions?</p>

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