A289209 Coefficients in expansion of E_4^3/E_6^2.

1, 1728, 1700352, 1332930816, 939690602496, 624182333927040, 399031077924476928, 248370528839869094400, 151578005556161702559744, 91116938989182168182098368, 54119528875319902426524072960, 31833210323194251819350736777984

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..365

Formula

G.f.: 1 + 1728 * q * Product_{k>=1} (1-q^k)^24 / E_6^2.

G.f.: (E_4*E_8)/(E_6*E_6) = (E_8*E_8)/(E_6*E_10). - Seiichi Manyama, Jun 29 2017

a(n) = 1728 * A289417(n - 1) for n > 0. - Seiichi Manyama, Jul 08 2017

a(n) ~ c * exp(2*Pi*n) * n, where c = 256 * Pi^6 / (3 * Gamma(1/4)^8) = 2.747700206704861755142526128354171788550012833617513654955480535522... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018

a(0) = 1, a(n) = (288/n)*Sum_{k=1..n} A300025(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 26 2018

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, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Crossrefs

(E_4^3/E_6^2)^(k/288): A289365 (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), this sequence (k=288).

Cf. A004009 (E_4), A013973 (E_6), A289063, A289210, A289417, A300025.

E_{k+2}/E_k: A288261 (k=4, 8), A288840 (k=6).

Sequence in context: A002519 A052068 A289210 * A114767 A350384 A165134

Adjacent sequences: A289206 A289207 A289208 * A289210 A289211 A289212

Keyword

nonn

Author

Seiichi Manyama, Jun 28 2017

Status

approved