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A264026 Expansion of (f(x^3) / f(x))^6 in powers of x where f() is a Ramanujan theta function.

%I A264026 #14 Feb 16 2025 08:33:27
%S A264026 1,-6,27,-92,279,-756,1913,-4536,10260,-22220,46479,-94176,185749,
%T A264026 -357426,673056,-1242404,2252772,-4017816,7058609,-12228060,20911230,
%U A264026 -35330324,59023728,-97568712,159693831,-258941124,416181510,-663337512,1048935414,-1646245836
%N A264026 Expansion of (f(x^3) / f(x))^6 in powers of x where f() is a Ramanujan theta function.
%C A264026 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A264026 G. C. Greubel, <a href="/A264026/b264026.txt">Table of n, a(n) for n = 0..2500</a>
%H A264026 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A264026 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A264026 Expansion of q^(-1/2) * (eta(q) * eta(q^4) * eta(q^6)^3 / (eta(q^2)^3 * eta(q^3) * eta(q^12)))^6 in powers of q.
%F A264026 Euler transform of period 12 sequence [ -6, 12, 0, 6, -6, 0, -6, 6, 0, 12, -6, 0, ...].
%F A264026 G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = (1/27) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132107.
%F A264026 Convolution inverse of A132107.
%F A264026 a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(13/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018
%e A264026 G.f. = 1 - 6*x + 27*x^2 - 92*x^3 + 279*x^4 - 756*x^5 + 1913*x^6 - 4536*x^7 + ...
%e A264026 G.f. = q - 6*q^3 + 27*q^5 - 92*q^7 + 279*q^9 - 756*q^11 + 1913*q^13 - 4536*q^15 + ...
%t A264026 a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3] / QPochhammer[ -x])^6, {x, 0, n}];
%t A264026 eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/2)* (eta[q]*eta[q^4]*eta[q^6]^3/(eta[q^2]^3*eta[q^3]*eta[q^12]))^6, {q, 0, 60}], q]; Table[a[[n]], {n,1,50}] (* _G. C. Greubel_, Jul 04 2018 *)
%o A264026 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3 / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A)))^6, n))};
%Y A264026 Cf. A000726, A132107.
%K A264026 sign
%O A264026 0,2
%A A264026 _Michael Somos_, Nov 01 2015