A191901 Number of compositions of odd natural numbers into 6 parts <= n.
0, 32, 364, 2048, 7812, 23328, 58824, 131072, 265720, 500000, 885780, 1492992, 2413404, 3764768, 5695312, 8388608, 12068784, 17006112, 23522940, 32000000, 42883060, 56689952, 74017944, 95551488, 122070312, 154457888, 193710244, 240945152, 297411660, 364500000, 443751840
Offset: 0
Examples
a(1)=32 compositions of odd numbers into 6 parts <=1.
1:(0,0,0,0,0,1)-->6!/(5!1!)= 6
3:(0,0,0,1,1,1)-->6!/(3!3!)=20
5:(0,1,1,1,1,1)-->6!/(1!5!)= 6
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32
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..10000
- Adi Dani, Restricted compositions of natural numbers
- Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1)
Programs
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Magma
[((n + 1)^6 - (1 + (-1)^n)/2)/2 : n in [0..25]]; // Vincenzo Librandi, Jul 03 2011
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Maple
A191901:=n->((n + 1)^6 - (1 + (-1)^n)/2)/2: seq(A191901(n), n=0..50); # Wesley Ivan Hurt, Apr 10 2017
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Mathematica
Table[Floor[1/2*((n + 1)^6 - (1 + (-1)^n)/2)], {n, 0, 30}] LinearRecurrence[{6,-14,14,0,-14,14,-6,1},{0,32,364,2048,7812,23328,58824,131072},40] (* Harvey P. Dale, Apr 12 2015 *)
Formula
a(n) = ((n + 1)^6 - (1 + (-1)^n)/2)/2.
G.f. -4*x*(8+43*x+78*x^2+43*x^3+8*x^4) / ( (1+x)*(x-1)^7 ). - R. J. Mathar, Jun 26 2011
a(0)=0, a(1)=32, a(2)=364, a(3)=2048, a(4)=7812, a(5)=23328, a(6)=58824, a(7)=131072, a(n)=6*a(n-1)-14*a(n-2)+14*a(n-3)-14*a(n-5)+ 14*a(n-6)- 6*a(n-7)+a(n-8). - Harvey P. Dale, Apr 12 2015