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A257920 Expansion of phi(x) * psi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A257920 #16 Jul 18 2025 05:21:23
%S A257920 1,2,0,1,4,0,0,2,0,3,2,0,2,2,0,0,2,0,3,4,0,0,2,0,0,4,0,2,2,0,1,2,0,0,
%T A257920 6,0,2,0,0,4,0,0,0,2,0,3,4,0,0,4,0,0,2,0,4,2,0,0,2,0,0,2,0,1,4,0,2,6,
%U A257920 0,0,2,0,2,2,0,0,0,0,0,4,0,4,2,0,3,2,0
%N A257920 Expansion of phi(x) * psi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A257920 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A257920 G. C. Greubel, <a href="/A257920/b257920.txt">Table of n, a(n) for n = 0..1000</a>
%H A257920 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A257920 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A257920 Expansion of q^(-3/8) * eta(q^2)^5 * eta(q^6)^2 / (eta(q)^2 * eta(q^3) * eta(q^4)^2) in powers of q.
%F A257920 Euler transform of period 12 sequence [ 2, -3, 3, -1, 2, -4, 2, -1, 3, -3, 2, -2, ...].
%F A257920 a(n) = A129402(4*n + 1) = A134177(4*n + 1) = A000377(8*n + 3) = A192013(8*n + 3).
%F A257920 a(3*n + 2) = 0. a(3*n + 1) = 2 * A128591(n).
%e A257920 G.f. = 1 + 2*x + x^3 + 4*x^4 + 2*x^7 + 3*x^9 + 2*x^10 + 2*x^12 + 2*x^13 + ...
%e A257920 G.f. = q^3 + 2*q^11 + q^27 + 4*q^35 + 2*q^59 + 3*q^75 + 2*q^83 + 2*q^99 + ...
%t A257920 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}];
%o A257920 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^6 + A)^2 / (eta(x + A)^2 * eta(x^3 + A) * eta(x^4 + A)^2), n))};
%Y A257920 Cf. A000377, A128591, A129402, A134177, A192013.
%K A257920 nonn
%O A257920 0,2
%A A257920 _Michael Somos_, Jul 12 2015