A290272 - OEIS

%I #25 May 07 2023 12:13:02

%S 1,750,201375,22695250,998651625,26031517500,480182965250,

%T 6889530585750,81442044063750,824111047734000,7333504889261250,

%U 58541361200675250,425628799655493875,2852238724568034000,17785782442113552000,104010815310940347500

%N 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).

%H Seiichi Manyama, <a href="/A290272/b290272.txt">Table of n, a(n) for n = 0..1000</a>

%H Steven R. Finch, <a href="/A000521/a000521_1.pdf">Modular forms on SL_2(Z)</a>, December 28, 2005. [Cached copy, with permission of the author]

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

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

%F 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

%F 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

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

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jul 25 2017