A357836 a(n) = Sum_{k=0..floor((n-2)/3)} Stirling1(n,3*k+2).

0, 0, 1, -3, 11, -49, 259, -1589, 11109, -87171, 758121, -7229859, 74905467, -836159961, 9980000667, -126422745813, 1686902233653, -23512989735963, 338917341235473, -4982536435536387, 73087736506615467, -1025163078325255233, 12286912220375608179

Offset

0,4

Links

Table of n, a(n) for n=0..22.

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Formula

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w*exp(w*x) + w^2*exp(w^2*x))/3 = x^2/2! + x^5/5! + x^8/8! + ... . Then the e.g.f. for the sequence is F(log(1+x)).

a(n) = (-1)^n * ( (-1)_n + w * (-w)_n + w^2 * (-w^2)_n )/3, where (x)_n is the Pochhammer symbol.

Prog

(PARI) a(n) = sum(k=0, (n-2)\3, stirling(n, 3*k+2, 1));
(PARI) my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(sum(k=0, N\3, log(1+x)^(3*k+2)/(3*k+2)!))))
(PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(w=(-1+sqrt(3)*I)/2); (-1)^n*round(Pochhammer(-1, n)+w*Pochhammer(-w, n)+w^2*Pochhammer(-w^2, n))/3;

Crossrefs

Cf. A357834, A357835.

Sequence in context: A340813 A004211 A180869 * A356291 A001339 A307030

Adjacent sequences: A357833 A357834 A357835 * A357837 A357838 A357839

Keyword

sign

Author

Seiichi Manyama, Oct 14 2022

Status

approved