A123187 - OEIS

A123187

Triangle of coefficients in expansion of (1 + 13*x)^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, 135150652, 501988136, 815730721

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OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..44.

Bulat Khaidarov, Enumeration of Certain k-Block Partitions of Barry, J. Int. Seq. 29 (2026), Art. 26.2.8. See p. 6 (Table 2).

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

EXAMPLE

Triangle begins:

1, 13;

1, 26, 169;

1, 39, 507, 2197;

1, 52, 1014, 8788, 28561;

1, 65, 1690, 21970, 142805, 371293;

...

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]

CROSSREFS

Cf. A001023, A013609, A013610, A013611, A013621, A038220, A038222.

Sequence in context: A094709 A236231 A040181 * A272797 A046733 A357312

Adjacent sequences: A123184 A123185 A123186 * A123188 A123189 A123190

KEYWORD

nonn,tabl,changed

AUTHOR

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

STATUS

approved