A289298 - OEIS

%I #24 Mar 06 2018 11:22:03

%S 1,93,-5661,741532,-113207799,19015433748,-3390166183729,

%T 629581913929419,-120437982238038210,23564574046009042869,

%U -4692899968498921291530,948024211601180444075739,-193775768073341380441728322

%N Expansion of (q*j(q))^(1/8) where j(q) is the elliptic modular invariant (A000521).

%H Seiichi Manyama, <a href="/A289298/b289298.txt">Table of n, a(n) for n = 0..424</a>

%F G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/8).

%F a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(11/8), where c = 0.2541876595230750963327533839122695596555059904123327336821622582369... = 3^(11/8) * sqrt(2 + sqrt(2)) * Gamma(1/3)^(9/4) * Gamma(3/8) / (2^(35/8) * exp(sqrt(3) * Pi/8) * Pi^(5/2)). - _Vaclav Kotesovec_, Jul 03 2017, updated Mar 06 2018

%F a(n) * A299827(n) ~ -3*2^(1/4)*sqrt(1+sqrt(2)) * exp(2*sqrt(3)*Pi*n) / (16*Pi*n^2). - _Vaclav Kotesovec_, Feb 20 2018

%t CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(3/8) / (2*QPochhammer[-1, x])^3, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 23 2017 *)

%t (q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/8) + O[q]^13 // CoefficientList[#, q]& (* _Jean-François Alcover_, Nov 02 2017 *)

%Y (q*j(q))^(k/24): A106205 (k=1), A289297 (k=2), this sequence (k=3), A289299 (k=4), A289300 (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).

%Y Cf. A000521, A192731.

%K sign

%O 0,2

%A _Seiichi Manyama_, Jul 02 2017