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Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^k )^(1/(1-x)).
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%I #18 Feb 06 2023 13:25:41

%S 1,1,8,63,644,7610,107994,1713726,30671024,603160344,12974475240,

%T 301879678320,7561610279112,202437968475288,5769455216675136,

%U 174234738889383480,5556311629901103360,186482786151757707840,6568881383985687359424,242221409390815100812224

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

%F a(0) = 1; a(n) = Sum_{k=1..n} A356298(k) * binomial(n-1,k-1) * a(n-k).

%t With[{nn=20},CoefficientList[Series[Product[1/((1-x^k)^k)^(1/(1-x)),{k,nn}],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Feb 06 2023 *)

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

%o (PARI) a356298(n) = n!*sum(k=1, n, sigma(k, 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, a356298(j)*binomial(i-1, j-1)*v[i-j+1])); v;

%Y Cf. A000219, A001157, A356298, A356335, A356336.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 04 2022