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A262930 Expansion of (psi(-q) / f(q^3))^2 in powers of q where psi(), f() are Ramanujan theta functions.

%I A262930 #12 Feb 16 2025 08:33:27
%S A262930 1,-2,1,-4,6,-2,12,-16,5,-28,36,-12,60,-76,24,-120,150,-46,228,-280,
%T A262930 86,-416,504,-152,732,-878,262,-1252,1488,-442,2088,-2464,725,-3408,
%U A262930 3996,-1168,5460,-6364,1852,-8600,9972,-2886,13344,-15400,4436,-20424,23472
%N A262930 Expansion of (psi(-q) / f(q^3))^2 in powers of q where psi(), f() are Ramanujan theta functions.
%C A262930 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262930 Seiichi Manyama, <a href="/A262930/b262930.txt">Table of n, a(n) for n = 0..10000</a>
%H A262930 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262930 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262930 Expansion of ( eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / ( eta(q^2) * eta(q^6)^3 ))^2 in powers of q.
%F A262930 Euler transform of period 12 sequence [ -2, 0, -4, -2, -2, 4, -2, -2, -4, 0, -2, 0, ...].
%F A262930 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A261369.
%F A262930 a(3*n) = A261320(n). a(3*n + 1) = -2 * A261325(n). a(3*n + 2) = A261369(n).
%F A262930 Convolution square of A139136.
%F A262930 a(2*n) = A263538(n). a(2*n + 1) = -2 * A263528(n).
%e A262930 G.f. = 1 - 2*x + x^2 - 4*x^3 + 6*x^4 - 2*x^5 + 12*x^6 - 16*x^7 + 5*x^8 + ...
%t A262930 a[ n_] := SeriesCoefficient[ (1/2) q^(-1/4) (EllipticTheta[ 2, Pi/4, q^(1/2)] / QPochhammer[ -q^3])^2, {q, 0, n}];
%o A262930 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / ( eta(x^2 + A) * eta(x^6 + A)^3 ))^2, n))};
%Y A262930 Cf. A139136, A261325, A261328, A261369, A263528, A263538.
%K A262930 sign
%O A262930 0,2
%A A262930 _Michael Somos_, Oct 04 2015