A357833 a(n) = Sum_{k=0..floor((n-2)/3)} 2^k * |Stirling1(n,3*k+2)|.

0, 0, 1, 3, 11, 52, 304, 2114, 16992, 154626, 1568706, 17535108, 213965520, 2828584824, 40259041188, 613656673476, 9971942784132, 172071391424832, 3141974627361216, 60523400730707208, 1226519845766281008, 26084378634267048984, 580854626450078463000

Offset: 0

Author

Seiichi Manyama, Oct 14 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A357831, A357832.

Programs

PARI

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

PARI

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

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

Formula

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w*exp(w*x) + w^2*exp(w^2*x))/3 = x^2/2! + x^5/5! + x^8/8! + ... . Then the e.g.f. for the sequence is F(-2^(1/3) * log(1-x))/(2^(2/3)).

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