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Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^(1/k) )^x.
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%I #65 Jan 06 2023 09:14:23

%S 1,0,2,6,28,210,1248,13020,102128,1248912,13457880,176726880,

%T 2362784928,36609693120,551337892896,9588702417840,171779733546240,

%U 3230529997766400,64714946343904512,1371420774325866240,29953522454811096960,698447624328756610560

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

%H Vaclav Kotesovec, <a href="/A355064/b355064.txt">Table of n, a(n) for n = 0..445</a>

%F a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * sigma_0(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).

%t a[0] := a[0] = 1; a[1] := a[1] = 0;

%t a[n_] := a[n] = Sum[Factorial[k]*DivisorSigma[0, k - 1]/(k - 1)*Binomial[n - 1, k - 1]* a[n - k], {k, 2, n}];

%t Table[a[n], {n, 0, 50}] (* _Sidney Cadot_, Jan 05 2023 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^(1/k))^x))

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*sigma(j-1, 0)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

%Y Cf. A354623, A356554.

%Y Cf. A000005, A066166, A356336, A356564.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 12 2022