A357835 a(n) = Sum_{k=0..floor((n-1)/3)} Stirling1(n,3*k+1).

0, 1, -1, 2, -5, 14, -35, -14, 1701, -26418, 351351, -4622982, 62705643, -890078826, 13297263525, -209438953542, 3477446002485, -60803484275898, 1117975706702127, -21580455768575886, 436591651807054107, -9241512424454751714, 204338436416329792941

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^2*exp(w*x) + w*exp(w^2*x))/3 = x + x^4/4! + x^7/7! + ... . Then the e.g.f. for the sequence is F(log(1+x)).

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

Prog

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

Crossrefs

Cf. A357834, A357836.

Sequence in context: A344236 A374699 A265226 * A369591 A299164 A131408

Adjacent sequences: A357832 A357833 A357834 * A357836 A357837 A357838

Keyword

sign

Author

Seiichi Manyama, Oct 14 2022

Status

approved