A356362 a(n) = Sum_{k=0..floor(n/3)} n^k * Stirling1(n,3*k).

1, 0, 0, 3, -24, 175, -1314, 10339, -84448, 696429, -5444700, 32897601, 53444304, -8071238721, 235927045536, -5630771421765, 126525509087232, -2799633511755963, 62154971516786616, -1396560425289392007, 31880150704745078400, -740188445913015688953

Offset

0,4

Links

Table of n, a(n) for n=0..21.

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Formula

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . a(n) = n! * [x^n] F(n^(1/3) * log(1+x)).

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

Prog

(PARI) a(n) = sum(k=0, n\3, n^k*stirling(n, 3*k, 1));
(PARI) a(n) = n!*polcoef(sum(k=0, n\3, n^k*log(1+x+x*O(x^n))^(3*k)/(3*k)!), n);
(PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(v=n^(1/3), w=(-1+sqrt(3)*I)/2); (-1)^n*round(Pochhammer(-v, n)+Pochhammer(-v*w, n)+Pochhammer(-v*w^2, n))/3;

Crossrefs

Cf. A356361, A356363.

Cf. A357834.

Sequence in context: A089697 A120741 A361553 * A356361 A292293 A073985

Adjacent sequences: A356359 A356360 A356361 * A356363 A356364 A356365

Keyword

sign

Author

Seiichi Manyama, Oct 16 2022

Status

approved