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A289063 Coefficients in expansion of E_6^2/Product_{k>=1} (1-q^k)^24.

%I A289063 #35 Feb 16 2025 08:33:48
%S A289063 1,-984,196884,21493760,864299970,20245856256,333202640600,
%T A289063 4252023300096,44656994071935,401490886656000,3176440229784420,
%U A289063 22567393309593600,146211911499519294,874313719685775360,4872010111798142520,25497827389410525184,126142916465781843075
%N A289063 Coefficients in expansion of E_6^2/Product_{k>=1} (1-q^k)^24.
%C A289063 Convolution square of A007242. - _Michael Somos_, Mar 31 2019
%H A289063 Vaclav Kotesovec, <a href="/A289063/b289063.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Seiichi Manyama)
%H A289063 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/j-Function.html">j-Function</a>
%H A289063 Wikipedia, <a href="https://en.wikipedia.org/wiki/J-invariant">j-invariant</a>
%F A289063 G.f.: Product_{k>=1} (1-q^k)^A289061(k).
%F A289063 a(n) = A000521(n-1) for n = 0 and n > 1.
%F A289063 a(n) ~ exp(4*Pi*sqrt(n)) / (sqrt(2) * n^(3/4)). - _Vaclav Kotesovec_, Jul 09 2017
%F A289063 G.f.: q * (j(q) - 1728) where j(q) is a modular function. - _Michael Somos_, Mar 31 2019
%e A289063 G.f. = (1-q)^984 * (1-q^2)^286752 * (1-q^3)^102360024 * ...
%e A289063 G.f. = 1 - 984*q + 196884*q^2 + 21493760*q^3 + 864299970*q^4 + 20245856256*q^5 + ... .
%t A289063 nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2 / Product[(1 - x^k)^24, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 09 2017 *)
%t A289063 a[ n_] := SeriesCoefficient[ q Series[ 1728 (KleinInvariantJ[Log[q] / (2 Pi I)] - 1), {q, 0, n}], {q, 0, n}]; (* _Michael Somos_, Mar 31 2019 *)
%o A289063 (PARI) {a(n) = my(A, U1, U2); if( n<0, 0, A = x * O(x^n); U1 = eta(x + A)^24; U2 = eta(x^2 + A)^24; polcoeff( (U1 - 512*x * U2)^2 * (U1 + 64*x * U2) / (U1^2 * U2), n))}; /* _Michael Somos_, Mar 31 2019 */
%Y A289063 Cf. A000594, A013973 (E_6), A289061, A289062.
%Y A289063 Cf. A000521 (j), A007242, A014708, A007240.
%K A289063 sign
%O A289063 0,2
%A A289063 _Seiichi Manyama_, Jun 23 2017