A289349 Coefficients in expansion of E_6^(11/12).

1, -462, -24948, -2518824, -654112074, -212483064024, -76819071738024, -29728723632736128, -12066341379893331300, -5073593348593538950566, -2192302482140061697816872, -968086916154014421082349304, -435126775136273350146250044888

Offset: 0

Author

Seiichi Manyama, Jul 03 2017

Keywords

• sign

Comments

In general, for 0 < m < 1, the expansion of (E_6)^m is asymptotic to -m * Gamma(1/4)^(16*m) * 3^(2*m) * exp(2*Pi*n) / (2^(13*m) * Pi^(12*m) * Gamma(1-m) * n^(1+m)). - Vaclav Kotesovec, Mar 05 2018

Crossrefs

E_6^(k/12): A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), this sequence (k=11).

Cf. A013973 (E_6), A288851.

Programs

Mathematica

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(11/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(11*A288851(n)/12).

a(n) ~ c * exp(2*Pi*n) / n^(23/12), where c = -11 * 2^(5/12) * 3^(5/6) * Pi^(11/3) / (128 * Gamma(1/12) * Gamma(3/4)^(44/3)) = -0.08406022472181281739983743854923746657261382508944840919197295490535... - Vaclav Kotesovec, Jul 08 2017