A290272 Expansion of j(q) * q * Product_{n>=1} ((1 - q^(5*n))/(1 - q^n))^6 where j(q) is the elliptic modular invariant (A000521).

1, 750, 201375, 22695250, 998651625, 26031517500, 480182965250, 6889530585750, 81442044063750, 824111047734000, 7333504889261250, 58541361200675250, 425628799655493875, 2852238724568034000, 17785782442113552000, 104010815310940347500

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]

Formula

Let b(q) = q * Product_{n>=1} ((1 - q^(5*n))/(1 - q^n))^6.

G.f.: j(q) * b(q) = (1 + 250*b(q) + 3125*b(q)^2)^3.

a(n) ~ exp(4*Pi*sqrt(6*n/5)) * 3^(1/4) / (2^(1/4) * 5^(13/4) * n^(3/4)). - Vaclav Kotesovec, Nov 10 2017

Empirical : Sum_{n>=0} a(n)/exp(2*Pi*n) = -3456/125+1728/125*5^(1/2), validated at 2048 digits. - Simon Plouffe, May 07 2023. Also equal to (12/(5*phi))^3, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 07 2023

Crossrefs

Cf. A000521, A121591 (b(q)).

Sequence in context: A237798 A015067 A129036 * A291524 A373206 A020391

Adjacent sequences: A290269 A290270 A290271 * A290273 A290274 A290275

Keyword

nonn

Author

Seiichi Manyama, Jul 25 2017

Status

approved