A357834 a(n) = Sum_{k=0..floor(n/3)} Stirling1(n,3*k).

1, 0, 0, 1, -6, 35, -224, 1603, -12810, 113589, -1109472, 11852841, -137611110, 1726238787, -23277264192, 335861699355, -5164348236138, 84316474011861, -1456893047937600, 26562992204112273, -509679388313669574, 10266675502780006947, -216625348636705401120

Offset: 0

Author

Seiichi Manyama, Oct 14 2022

Keywords

• sign

Links

• Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Crossrefs

Cf. A357835, A357836.

Cf. A105752.

Programs

PARI

a(n) = sum(k=0, n\3, stirling(n, 3*k, 1));

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N\3, log(1+x)^(3*k)/(3*k)!)))

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)+Pochhammer(-w, n)+Pochhammer(-w^2, n))/3;

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(log(1+x)).

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