A356394 Expansion of e.g.f. ( Product_{k>0} (1+x^k)^k )^(1/(1-x)).

1, 1, 6, 51, 452, 5210, 68514, 1032906, 17352320, 323948376, 6594052680, 145585638000, 3461441121192, 88092914635128, 2388119359650192, 68667743686492440, 2086307088847714560, 66762608893508354880, 2243693428523140377024, 78982154604162553529664

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..423

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356391(k) * binomial(n-1,k-1) * a(n-k).

Mathematica

nmax = 20; CoefficientList[Series[Product[(1+x^k)^k, {k, 1, nmax}]^(1/(1-x)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 07 2022 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^k)^(1/(1-x))))
(PARI) a356391(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356391(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Crossrefs

Cf. A356392, A356393.

Cf. A026007, A356337, A356391.

Sequence in context: A050916 A011790 A292840 * A202754 A363112 A180901

Adjacent sequences: A356391 A356392 A356393 * A356395 A356396 A356397

Keyword

nonn

Author

Seiichi Manyama, Aug 05 2022

Status

approved