A383838 Expansion of 1/((1-x) * (1-4*x) * (1-9*x) * (1-16*x)).

1, 30, 627, 11440, 196053, 3255330, 53157079, 860181300, 13850000505, 222384254950, 3565207699131, 57106865357880, 914281747641757, 14633655168987690, 234184807922193183, 3747373855152257980, 59961734043737254209, 959421515974412698350, 15351048197153778821635

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..830

Andrii Husiev, Extended Central Factorial Numbers and the Flickering Operator, arXiv:2605.06689 [math.GM], 2026. See pp. 3-4.

Index entries for linear recurrences with constant coefficients, signature (30,-273,820,-576).

Formula

a(n) = A269945(n+4,4).

a(n) = 30*a(n-1) - 273*a(n-2) + 820*a(n-3) - 576*a(n-4).

a(n) = (2*16^(n+3) - 9^(n+4) + 14*4^(n+3) - 7)/2520.

sinh(x)^8/8! = Sum_{k>=0} 4^k * a(k) * x^(2*k+8)/(2*k+8)!.

a(n) = (1/8!) * Sum_{k=0..8} (-1)^k * (4-k)^(2*n+8) * binomial(8,k).

a(n) = Sum_{k=0..2*n} (-4)^k * binomial(2*n+8,k) * Stirling2(2*n-k+8,8).

a(n) = Sum_{k=0..2*n} (-1)^k * Stirling2(k+4,4) * Stirling2(2*n-k+4,4).

Prog

(PARI) a(n) = (2*16^(n+3)-9^(n+4)+14*4^(n+3)-7)/2520;

Crossrefs

Cf. A002451, A269945.

Sequence in context: A285235 A075911 A001719 * A004359 A001777 A205828

Adjacent sequences: A383835 A383836 A383837 * A383839 A383840 A383841

Keyword

nonn,easy

Author

Seiichi Manyama, May 11 2025

Status

approved