A357832 a(n) = Sum_{k=0..floor((n-1)/3)} 2^k * |Stirling1(n,3*k+1)|.

0, 1, 1, 2, 8, 44, 290, 2194, 18690, 177072, 1848048, 21079332, 260998584, 3487438476, 50030096844, 767092681992, 12520306878720, 216760973139072, 3967857438205320, 76575231882844056, 1553981718941428824, 33082675130470434336, 737250032464248840192

Offset: 0

Author

Seiichi Manyama, Oct 14 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A357831, A357833.

Programs

Mathematica

a[n_] := With[{v = 2^(1/3), w = (-1 + Sqrt[3]*I)/2}, Round[(Pochhammer[v, n] + w^2*Pochhammer[v*w, n] + w*Pochhammer[v*w^2, n])/(3*v)]];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Oct 16 2022, after 3rd PARI code *)

PARI

a(n) = sum(k=0, (n-1)\3, 2^k*abs(stirling(n, 3*k+1, 1)));

PARI

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

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

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(-2^(1/3) * log(1-x))/(2^(1/3)).

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