A123187 Triangle of coefficients in expansion of (1+13x)^n.

1, 1, 13, 1, 26, 169, 1, 39, 507, 2197, 1, 52, 1014, 8788, 28561, 1, 65, 1690, 21970, 142805, 371293, 1, 78, 2535, 43940, 428415, 2227758, 4826809, 1, 91, 3549, 76895, 999635, 7797153, 33787663, 62748517, 1, 104, 4732, 123032, 1999270, 20792408

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Oct 03 2006

Keywords

• nonn
• tabl

Comments

T(n,k) equals the number of n-length words on {0,1,...,13} having n-k zeros. - Milan Janjic, Jul 24 2015

Examples

			1
1, 13
1, 26, 169
1, 39, 507, 2197
1, 52, 1014, 8788, 28561
1, 65, 1690, 21970, 142805, 371293
		

Crossrefs

Cf. A013609, A013610, A013611, A013621, A038220, A038222.

Programs

Maple

T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+13*x)^n):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 24 2015

Mathematica

p[0, x] = 1; p[1, x] = 13*x + 1; p[k_, x_] := p[k, x] = (13*x + 1)*p[k - 1, x]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]

Formula

p(k, x) = (13*x + 1)*p(k - 1, x).

T(n,k) = 13^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*12^(n-i). Row sums are 14^n = A001023. G.f.: 1 / [1 - x(1+13y)]. - Mircea Merca, Apr 28 2012