A289297 - OEIS

%I #24 Mar 06 2018 11:18:53

%S 1,62,-4735,651070,-103766140,17999397756,-3292567703035,

%T 624659270035130,-121698860487451255,24194029851560118900,

%U -4886913657541566648179,999849040331683393909232,-206741394604073327046805355

%N Expansion of (q*j(q))^(1/12) where j(q) is the elliptic modular invariant (A000521).

%H Seiichi Manyama, <a href="/A289297/b289297.txt">Table of n, a(n) for n = 0..424</a>

%F G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/12).

%F a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(5/4), where c = 0.200236163401945306105645017761063156355568043417672219092096121424... = 3^(1/4) * Gamma(1/4) * Gamma(1/3)^(3/2) / (2^(11/4) * exp(Pi/(4 * sqrt(3))) * Pi^2). - _Vaclav Kotesovec_, Jul 03 2017, updated Mar 06 2018

%F a(n) * A299826(n) ~ -exp(2*sqrt(3)*n*Pi) / (2^(5/2)*Pi*n^2). - _Vaclav Kotesovec_, Feb 20 2018

%t CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(1/4) / (2*QPochhammer[-1, x])^2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 23 2017 *)

%t (q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/12) + O[q]^13 //

%t CoefficientList[#, q]& (* _Jean-François Alcover_, Nov 02 2017 *)

%Y (q*j(q))^(k/24): A106205 (k=1), this sequence (k=2), A289298 (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