A383840 Expansion of 1/((1-x) * (1-4*x) * (1-9*x) * (1-16*x) * (1-25*x)).

1, 55, 2002, 61490, 1733303, 46587905, 1217854704, 31306548900, 796513723005, 20135227330075, 506945890951006, 12730754139133030, 319183135225967507, 7994212035818175365, 200089485703376577308, 5005984516439566690680, 125209574645032904521209

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..714

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 (55,-1023,7645,-21076,14400).

Formula

a(n) = A269945(n+5,5).

a(n) = 55*a(n-1) - 1023*a(n-2) + 7645*a(n-3) - 21076*a(n-4) + 14400*a(n-5).

a(n) = (5*25^(n+4) - 2*16^(n+5) + 9^(n+6) - 6*4^(n+6) + 42)/362880.

sinh(x)^10/10! = Sum_{k>=0} 4^k * a(k) * x^(2*k+10)/(2*k+10)!.

a(n) = (1/10!) * Sum_{k=0..10} (-1)^k * (5-k)^(2*n+10) * binomial(10,k).

a(n) = Sum_{k=0..2*n} (-5)^k * binomial(2*n+10,k) * Stirling2(2*n-k+10,10).

a(n) = Sum_{k=0..2*n} (-1)^k * Stirling2(k+5,5) * Stirling2(2*n-k+5,5).

Prog

(PARI) a(n) = (5*25^(n+4)-2*16^(n+5)+9^(n+6)-6*4^(n+6)+42)/362880;

Crossrefs

Cf. A269945, A383838.

Sequence in context: A217758 A346325 A383635 * A381584 A240687 A053113

Adjacent sequences: A383837 A383838 A383839 * A383841 A383842 A383843

Keyword

nonn,easy

Author

Seiichi Manyama, May 11 2025

Status

approved