A289454 - OEIS

%I #25 Mar 05 2018 07:49:09

%S 1,-2232,2730564,-2425008768,1748443340826,-1085940040502592,

%T 602376210735356376,-305671359557586479616,144309502321265349235035,

%U -64175062238369552680712096,27135987216939727366492175940,-10990160397215122310079248998656

%N Expansion of 1/j^3 where j is the elliptic modular invariant (A000521).

%H Seiichi Manyama, <a href="/A289454/b289454.txt">Table of n, a(n) for n = 3..419</a>

%F a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) * n^8, where c = (4*Pi^36) / (35 * 3^11 * Gamma(1/3)^54) = 0.00000000000395425888452699792549199102489774693818147819519... - _Vaclav Kotesovec_, Jul 07 2017, updated Mar 05 2018

%t nmax = 20; Drop[CoefficientList[Series[((1 - (1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}])^2/(1 + 240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}])^3)/1728)^3, {x, 0, nmax}], x], 3] (* _Vaclav Kotesovec_, Jul 07 2017 *)

%t a[n_] := SeriesCoefficient[1/(1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^3, {q, 0, n}]; Table[a[n], {n, 3, 14}] (* _Jean-François Alcover_, Nov 02 2017 *)

%Y Cf. A000521 (j).

%Y 1/j^k: A066395 (k=1), A288727 (k=2), this sequence (k=3), A289455 (k=4), A289512 (k=5), A289513(k=6), A289514 (k=7), A289515 (k=8), A289516 (k=9), A289517 (k=10).

%K sign

%O 3,2

%A _Seiichi Manyama_, Jul 06 2017