A191496 -id:A191496 - cp's OEIS frontend

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

Adi Dani, Jun 11 2011

nonn

			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
  ------------------------------------------
                                         256

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

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

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

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

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

A191687 Table T(n,k) = ceiling((1/2)*((k+1)^n+(1+(-1)^k)/2)) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 14, 8, 3, 1, 1, 16, 41, 32, 13, 3, 1, 1, 32, 122, 128, 63, 18, 4, 1, 1, 64, 365, 512, 313, 108, 25, 4, 1, 1, 128, 1094, 2048, 1563, 648, 172, 32, 5, 1

Offset: 1

Adi Dani, Jun 11 2011

T(n,k) is the number of compositions of even natural numbers into n parts <= k.

			Top left corner:
  1, 1, 1,  1,  1,...
  1, 1, 2,  2,  3,...
  1, 2, 5,  8, 13,...
  1, 4,14, 32, 63,...
  1, 8,41,128,313,...
T(2,4)=13: there are 13 compositions of even natural numbers into 2 parts <=4
0: (0,0);
2: (0,2), (2,0), (1,1);
4: (0,4), (4,0), (1,3), (3,1), (2,2);
6: (2,4), (4,2), (3,3);
8: (4,4).

Adi Dani, Restricted compositions of natural numbers

Rows sums gives: A000012, A004526, A000982, A036486, A171714, A191484, A191489, A191494, A191495, A191496

Columns sums gives: A000012, A000079, A007051, A004171, A034478, A081341, A034494, A092811, A083884, A093143

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

cp's OEIS Frontend

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

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