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A225912 Expansion of q * (phi(-q^2) * psi(-q)^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions.

%I A225912 #21 Feb 16 2025 08:33:19
%S A225912 0,1,-8,20,0,-74,96,-24,0,157,-432,124,0,478,704,-1480,0,-1198,792,
%T A225912 3044,0,-480,-4320,184,0,2351,3344,-1720,0,-3282,5184,-5728,0,2480,
%U A225912 -4752,1776,0,10326,-6688,9560,0,-8886,-8448,-9188,0,-11618,32832,23664,0,-16231
%N A225912 Expansion of q * (phi(-q^2) * psi(-q)^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions.
%C A225912 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A225912 G. C. Greubel, <a href="/A225912/b225912.txt">Table of n, a(n) for n = 0..2500</a>
%H A225912 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A225912 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A225912 Expansion of (eta(q)^2 * eta(q^4))^4 in powers of q.
%F A225912 Euler transform of period 4 sequence [-8, -8, -8, -12, ...].
%F A225912 G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^14 (t/i)^6 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A225872.
%F A225912 G.f.: x * (Product_{k>0} (1 - x^k)^2 * (1 - x^(4*k)))^4.
%e A225912 G.f. = q - 8*q^2 + 20*q^3 - 74*q^5 + 96*q^6 - 24*q^7 + 157*q^9 - 432*q^10 + ...
%t A225912 a[ n_] := SeriesCoefficient[ -(EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, I q^(1/2)]^2 / 4 )^4, {q, 0, n}];
%t A225912 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q]^2 QPochhammer[ q^4])^4, {q, 0, n}];
%o A225912 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A))^4, n))};
%Y A225912 Cf. A225923 (bisection?)
%K A225912 sign
%O A225912 0,3
%A A225912 _Michael Somos_, May 20 2013