A383701 Coefficient of x^3 in expansion of (x+1) * (x+5) * ... * (x+4*n-3).

0, 0, 0, 1, 28, 730, 20460, 633619, 21740040, 823020596, 34174098440, 1546855384261, 75883563554436, 4013184755214414, 227719025845257492, 13804358188086757719, 890571834923460488784, 60933371174617735181160, 4407783770975985847999440, 336154167664942342604334345

Offset: 0

Author

Seiichi Manyama, May 06 2025

Keywords

• nonn

Crossrefs

Column k=3 of A290319.

Programs

PARI

a(n) = polcoef(prod(k=0, n-1, x+4*k+1), 3);

Formula

a(n) = Sum_{k=3..n} 4^(n-k) * binomial(k,3) * |Stirling1(n,k)|.

a(n) = Sum_{k=3..n} (4*n-3)^(k-3) * 4^(n-k) * binomial(k,3) * Stirling1(n,k).

E.g.f.: f(x) * log(f(x))^3 / 6, where f(x) = 1/(1 - 4*x)^(1/4).

Conjecture D-finite with recurrence a(n) +4*(-4*n+9)*a(n-1) +2*(48*n^2-264*n+371)*a(n-2) -4*(4*n-13)*(16*n^2-104*n+173)*a(n-3) +(4*n-15)^4*a(n-4)=0. - R. J. Mathar, May 07 2025