A289365 Coefficients in expansion of (E_4^3/E_6^2)^(1/288).

1, 6, 738, 402444, 138030342, 63625789080, 27583809566796, 12841110779519280, 5988752245273028886, 2859827345620916000346, 1377856546809576262931880, 671500179383482897207038108, 329754232921005442388958831684

Offset: 0

Author

Seiichi Manyama, Jul 04 2017

Keywords

• nonn

Comments

In general, for m > 0, the expansion of (E_4^3/E_6^2)^m is asymptotic to 2^(8*m) * Pi^(6*m) * exp(2*Pi*n) / (3^m * Gamma(1/4)^(8*m) * Gamma(2*m) * n^(1-2*m)). - Vaclav Kotesovec, Mar 04 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..368

Crossrefs

(E_4^3/E_6^2)^(k/288): this sequence (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), A299698 (k=6), A299943 (k=8), A299949 (k=9), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

Cf. A289209 (E_4^3/E_6^2), A289366, A289367, A300025.

Programs

Mathematica

nmax = 20; CoefficientList[Series[((1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3 / (1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2)^(1/288), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A289367(n)).

a(n) ~ c * exp(2*Pi*n) / n^(143/144), where c = 2^(1/36) * Pi^(1/48) / (3^(1/288) * Gamma(1/144) * Gamma(1/4)^(1/36)) = 0.00699657322237604876174085217217686... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A300025(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 25 2018

a(n) * A289366(n) ~ -sin(Pi/144) * exp(4*Pi*n) / (144*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018