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

0,5

Links

Robert Israel, Table of n, a(n) for n = 0..365

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

Conjecture confirmed using differential equation y + (624*x - 156)*y' + (2656*x^2 - 1328*x + 166)*y'' + (1792*x^3 - 1344*x^2 + 336*x - 28)*y''' + (256*x^4 - 256*x^3 + 96*x^2 - 16*x + 1)*y'''' satisfied by the e.g.f. - Robert Israel, Mar 12 2026

Maple

f:= gfun:-rectoproc({(4*n + 1)^4*a(n) - 4*a(n + 1)*(4*n + 3)*(16*n^2 + 24*n + 13) + 2*(48*n^2 + 120*n + 83)*a(n + 2) - 4*a(n + 3)*(4*n + 7) + a(n + 4), a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1}, a(n), remember):
map(f, [$0..20]); # Robert Israel, Mar 12 2026

Prog

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

Crossrefs

Column k=3 of A290319.

Sequence in context: A012808 A226991 A277060 * A229463 A097834 A162830

Adjacent sequences: A383698 A383699 A383700 * A383702 A383703 A383704

Keyword

nonn

Author

Seiichi Manyama, May 06 2025

Status

approved