cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A318579 Expansion of Product_{i>=1, j>=1} ((1 + x^(i*j))/(1 - x^(i*j)))^(i*j).

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%I A318579 #12 Apr 03 2019 02:56:26
%S A318579 1,2,10,30,98,270,786,2046,5418,13556,33726,81002,192902,447562,
%T A318579 1027990,2316750,5165398,11345298,24668952,52972902,112688802,
%U A318579 237193354,494933514,1023238806,2098662698,4269141516,8620916966,17280687472,34405835066,68044209950,133732805458
%N A318579 Expansion of Product_{i>=1, j>=1} ((1 + x^(i*j))/(1 - x^(i*j)))^(i*j).
%C A318579 Convolution of A280540 and A280541.
%H A318579 Vaclav Kotesovec, <a href="/A318579/b318579.txt">Table of n, a(n) for n = 0..10000</a>
%F A318579 G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*tau(k)), where tau(k) = number of divisors of k (A000005).
%F A318579 G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (1 - (-1)^(k/d))*d^2*tau(d) ) * x^k/k).
%F A318579 log(a(n)) ~ 3^(2/3) * (7*Zeta(3))^(1/3) * log(n)^(1/3) * n^(2/3) / 2^(4/3). - _Vaclav Kotesovec_, Sep 03 2018
%p A318579 a:=series(mul(mul(((1+x^(i*j))/(1-x^(i*j)))^(i*j),j=1..100),i=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # _Paolo P. Lava_, Apr 02 2019
%t A318579 nmax = 30; CoefficientList[Series[Product[Product[((1 + x^(i j))/(1 - x^(i j)))^(i j), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x]
%t A318579 nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k DivisorSigma[0, k]), {k, 1, nmax}], {x, 0, nmax}], x]
%t A318579 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(1 - (-1)^(k/d)) d^2 DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 30}]
%Y A318579 Cf. A000005, A038040, A156616, A280540, A280541, A301554, A301555.
%K A318579 nonn
%O A318579 0,2
%A A318579 _Ilya Gutkovskiy_, Aug 29 2018