A289366 Coefficients in expansion of (E_6^2/E_4^3)^(1/288).

1, -6, -702, -393804, -132734778, -61428055320, -26480146877172, -12318952616296752, -5730786812846192490, -2732960583228848850522, -1314627022075990658598360, -639871947654492158944455132, -313833506047227501170833823292

Offset

0,2

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..367

Formula

G.f.: (1 - 1728/j)^(1/288).

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

a(n) ~ c * exp(2*Pi*n) / n^(145/144), where c = -Gamma(1/4)^(1/36) / (48 * 2^(1/36) * 3^(287/288) * Pi^(1/48) * Gamma(143/144)) = -0.006892157290355982837398273285864980110980721215574657372422958228077... - 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) * A289365(n) ~ -sin(Pi/144) * exp(4*Pi*n) / (144*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

Mathematica

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

Crossrefs

(E_6^2/E_4^3)^(k/288): this sequence (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).

Cf. A000521 (j), A066395 (1/j), A289365, A289367, A300025, A305886.

Sequence in context: A343140 A116298 A112637 * A253120 A354250 A218559

Adjacent sequences: A289363 A289364 A289365 * A289367 A289368 A289369

Keyword

sign

Author

Seiichi Manyama, Jul 04 2017

Status

approved