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A326831 Expansion of Product_{i>=2, j>=2} (1 + x^(i*j))^j.

%I A326831 #12 Oct 26 2019 11:21:37
%S A326831 1,0,0,0,2,0,5,0,7,3,17,0,37,6,58,23,120,21,235,67,390,161,726,230,
%T A326831 1349,521,2225,1055,3990,1714,7040,3341,11604,6294,20053,10500,34252,
%U A326831 19115,56055,34168,94306,56998,157078,99515,254766,171484,419287,283565
%N A326831 Expansion of Product_{i>=2, j>=2} (1 + x^(i*j))^j.
%C A326831 Weigh transform of A048050.
%C A326831 Convolution of A326831 and A025147 is A319107. - _Vaclav Kotesovec_, Oct 26 2019
%H A326831 Vaclav Kotesovec, <a href="/A326831/b326831.txt">Table of n, a(n) for n = 0..2000</a>
%F A326831 G.f.: Product_{k>=1} (1 + x^k)^A048050(k).
%F A326831 a(n) ~ exp(3*(2*(Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3)/4 - Pi^2 * n^(1/3) / (2^(7/3) * ((Pi^2 - 6)*Zeta(3))^(1/3)) - Pi^4 / (96*(Pi^2 - 6)*Zeta(3))) * 2^(19/24) * ((Pi^2 - 6)*Zeta(3))^(1/6) / (sqrt(3*Pi) * n^(2/3)). - _Vaclav Kotesovec_, Oct 26 2019
%p A326831 with(numtheory):
%p A326831 g:= proc(n) option remember; `if`(n<4, 0, sigma(n)-1-n) end:
%p A326831 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A326831       add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))
%p A326831     end:
%p A326831 a:= n-> b(n$2):
%p A326831 seq(a(n), n=0..50);  # _Alois P. Heinz_, Oct 20 2019
%t A326831 nmax = 47; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[1, k] - k - 1), {k, 2, nmax}], {x, 0, nmax}], x]
%t A326831 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[d == 1, 0, (-1)^(k/d + 1) d (DivisorSigma[1, d] - d - 1)], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 47}]
%Y A326831 Cf. A048050, A192065, A319107, A326830.
%K A326831 nonn
%O A326831 0,5
%A A326831 _Ilya Gutkovskiy_, Oct 20 2019