A357831 a(n) = Sum_{k=0..floor(n/3)} 2^k * |Stirling1(n,3*k)|.

1, 0, 0, 2, 12, 70, 454, 3332, 27552, 254400, 2598852, 29125932, 355455468, 4693396656, 66671326176, 1013916648840, 16436063079552, 282920894841096, 5153797995148296, 99052313167391760, 2003040751641857856, 42513854724369719136, 944959706480298199824

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) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . Then the e.g.f. for the sequence is F(-2^(1/3) * log(1-x)).

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

Prog

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

Crossrefs

Cf. A357832, A357833.

Sequence in context: A059229 A001251 A143357 * A012426 A012421 A012381

Adjacent sequences: A357828 A357829 A357830 * A357832 A357833 A357834

Keyword

nonn

Author

Seiichi Manyama, Oct 14 2022

Status

approved