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Expansion of e.g.f. ( Product_{k>0} (1+x^k)^k )^(1/(1-x)).
6

%I #16 Aug 16 2022 10:16:33

%S 1,1,6,51,452,5210,68514,1032906,17352320,323948376,6594052680,

%T 145585638000,3461441121192,88092914635128,2388119359650192,

%U 68667743686492440,2086307088847714560,66762608893508354880,2243693428523140377024,78982154604162553529664

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

%H Seiichi Manyama, <a href="/A356394/b356394.txt">Table of n, a(n) for n = 0..423</a>

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

%t 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 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^k)^(1/(1-x))))

%o (PARI) a356391(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/k);

%o 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;

%Y Cf. A356392, A356393.

%Y Cf. A026007, A356337, A356391.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 05 2022