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A097243 Expansion of 1 + 32 * (eta(q^4) / eta(q))^8 in powers of q.

%I A097243 #17 Dec 08 2017 09:47:00
%S A097243 1,32,256,1408,6144,22976,76800,235264,671744,1809568,4640256,
%T A097243 11404416,27009024,61905088,137803776,298806528,632684544,1310891584,
%U A097243 2662655232,5310231424,10412576768,20098970624,38231811072,71734039808,132875747328,243175399136
%N A097243 Expansion of 1 + 32 * (eta(q^4) / eta(q))^8 in powers of q.
%C A097243 Expansion of a q-series used in construction of j(tau) to j(2tau) iteration.
%D A097243 H. Cohn, Introduction to the construction of class fields, Cambridge 1985, p. 191
%H A097243 G. C. Greubel, <a href="/A097243/b097243.txt">Table of n, a(n) for n = 0..1000</a>
%F A097243 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u+3)^2 - 8*(u+1)*v^2.
%F A097243 a(n) = 32*A092877(n), if n>0. a(n) = A007096(4*n).
%F A097243 a(n) = A014969(2*n) = A139820(2*n) = A189925(4*n) = A212318(4*n) = A232358(4*n). - _Michael Somos_, Dec 15 2016
%F A097243 G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 1/8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007248. - _Michael Somos_, Dec 15 2016
%F A097243 a(n) ~ exp(2*Pi*sqrt(n))/(16*n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2017
%e A097243 G.f. = 1 + 32*x + 256*x^2 + 1408*x^3 + 6144*x^4 + 22976*x^5 + 76800*x^6 + ...
%t A097243 a[ n_] := SeriesCoefficient[ 1 + 32 x (QPochhammer[ x^4] / QPochhammer[ x])^8, {x, 0, n}]; (* _Michael Somos_, Dec 15 2016 *)
%o A097243 (PARI) {a(n) = my(A); if( n<0, 0, A = x^n * O(x); polcoeff( 1 + 32 * x * (eta(x^4 + A) / eta(x + A))^8, n))};
%Y A097243 Cf. A007248, A007096, A014969, A092877, A139820, A189925, A212318, A232358.
%K A097243 nonn
%O A097243 0,2
%A A097243 _Michael Somos_, Aug 02 2004