A191680 Number of compositions of odd natural numbers into 9 parts <= n.

0, 256, 9841, 131072, 976562, 5038848, 20176803, 67108864, 193710244, 500000000, 1178973845, 2579890176, 5302249686, 10330523392, 19221679687, 34359738368, 59293938248, 99179645184, 161343848889, 256000000000, 397140023290, 603634608896, 900576330731, 1320903770112, 1907348632812, 2714751839488

Offset: 0

Author

Adi Dani, Jun 11 2011

Keywords

• nonn

Examples

			a(1)=256 compositions of odd numbers into 9 parts <= 1:
  1: (0,0,0,0,0,0,0,0,1) --> 9!/(8!1!) =   9
  3: (0,0,0,0,0,0,1,1,1) --> 9!/(6!3!) =  84
  5: (0,0,0,0,1,1,1,1,1) --> 9!/(4!5!) = 126
  7: (0,0,1,1,1,1,1,1,1) --> 9!/(2!7!) =  36
  9: (1,1,1,1,1,1,1,1,1) --> 9!/(0!9!) =   1
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                                         256
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Adi Dani, Restricted compositions of natural numbers

Programs

Magma

[1/2*((n + 1)^9 - (1 + (-1)^n)/2): n in [0..30]]; // Vincenzo Librandi, Jun 16 2011

Mathematica

Table[Floor[1/2*((n + 1)^9 - (1 + (-1)^n)/2)], {n, 0, 25}]

Formula

a(n) = ((n + 1)^9 - (1 + (-1)^n)/2)/2.

G.f.: x*(256 + 7537*x + 51463*x^2 + 122149*x^3 + 122275*x^4 + 51379*x^5 + 7573*x^6 + 247*x^7 + x^8) / ( (1+x)*(x-1)^10 ). - R. J. Mathar, Jun 29 2011

a(n) = 9*a(n-1) - 35*a(n-2) + 75*a(n-3) - 90*a(n-4) + 42*a(n-5) + 42*a(n-6) - 90*a(n-7) + 75*a(n-8) - 35*a(n-9) + 9*a(n-10) - a(n-11). - R. J. Mathar, Jun 29 2011

a(2n) = A191496(2n) - 1. a(2n+1) = A191496(2n+1). - R. J. Mathar, Jun 29 2011

Extensions

Offset changed from 1 to 0 by Vincenzo Librandi, Jun 16 2011