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A257536 Expansion of phi(-x^4)^2 * f(-x^1, -x^5) in powers of x where phi(), f() are Ramanujan theta functions.

%I A257536 #10 Feb 16 2025 08:33:25
%S A257536 1,-1,0,0,-4,3,0,0,5,0,0,0,-4,-4,0,0,9,-4,0,0,-12,3,0,0,8,12,0,0,-8,
%T A257536 -4,0,0,8,-5,0,0,-12,0,0,0,13,0,0,0,-8,-8,0,0,16,-4,0,0,-12,12,0,0,13,
%U A257536 12,0,0,-20,-8,0,0,8,-9,0,0,-16,12,0,0,16,0,0,0
%N A257536 Expansion of phi(-x^4)^2 * f(-x^1, -x^5) in powers of x where phi(), f() are Ramanujan theta functions.
%C A257536 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A257536 G. C. Greubel, <a href="/A257536/b257536.txt">Table of n, a(n) for n = 0..2500</a>
%H A257536 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A257536 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A257536 Expansion of q^(-1/3) * eta(q) * eta(q^4)^4 * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^8)^2) in powers of q.
%F A257536 Euler transform of period 24 sequence [ -1, 0, 0, -4, -1, -1, -1, -2, 0, 0, -1, -5, -1, 0, 0, -2, -1, -1, -1, -4, 0, 0, -1, -3, ...].
%F A257536 a(4*n + 2) = a(4*n + 3) = 0. 2 * a(n) = A116597(3*n + 1).
%e A257536 G.f. = 1 - x - 4*x^4 + 3*x^5 + 5*x^8 - 4*x^12 - 4*x^13 + 9*x^16 - 4*x^17 + ...
%e A257536 G.f. = q - q^4 - 4*q^13 + 3*q^16 + 5*q^25 - 4*q^37 - 4*q^40 + 9*q^49 + ...
%t A257536 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] EllipticTheta[ 4, 0, x^4]^2 EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}];
%o A257536 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^4 * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^8 + A)^2), n))};
%Y A257536 Cf. A116597.
%K A257536 sign
%O A257536 0,5
%A A257536 _Michael Somos_, Apr 28 2015