A289063 Coefficients in expansion of E_6^2/Product_{k>=1} (1-q^k)^24.

1, -984, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184, 126142916465781843075

Offset

0,2

Comments

Convolution square of A007242. - Michael Somos, Mar 31 2019

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

Eric Weisstein's World of Mathematics, j-Function

Wikipedia, j-invariant

Formula

G.f.: Product_{k>=1} (1-q^k)^A289061(k).

a(n) = A000521(n-1) for n = 0 and n > 1.

a(n) ~ exp(4*Pi*sqrt(n)) / (sqrt(2) * n^(3/4)). - Vaclav Kotesovec, Jul 09 2017

G.f.: q * (j(q) - 1728) where j(q) is a modular function. - Michael Somos, Mar 31 2019

Example

G.f. = (1-q)^984 * (1-q^2)^286752 * (1-q^3)^102360024 * ...
G.f. = 1 - 984*q + 196884*q^2 + 21493760*q^3 + 864299970*q^4 + 20245856256*q^5 + ... .

Mathematica

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}])^2 / Product[(1 - x^k)^24, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)
a[ n_] := SeriesCoefficient[ q Series[ 1728 (KleinInvariantJ[Log[q] / (2 Pi I)] - 1), {q, 0, n}], {q, 0, n}]; (* Michael Somos, Mar 31 2019 *)

Prog

(PARI) {a(n) = my(A, U1, U2); if( n<0, 0, A = x * O(x^n); U1 = eta(x + A)^24; U2 = eta(x^2 + A)^24; polcoeff( (U1 - 512*x * U2)^2 * (U1 + 64*x * U2) / (U1^2 * U2), n))}; /* Michael Somos, Mar 31 2019 */

Crossrefs

Cf. A000594, A013973 (E_6), A289061, A289062.

Cf. A000521 (j), A007242, A014708, A007240.

Sequence in context: A249201 A249323 A249216 * A289061 A294183 A288840

Adjacent sequences: A289060 A289061 A289062 * A289064 A289065 A289066

Keyword

sign

Author

Seiichi Manyama, Jun 23 2017

Status

approved