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A224822 Expansion of phi(-q) * phi(-q^3)^2 in powers of q where phi() is a Ramanujan theta function.

%I A224822 #16 Feb 16 2025 08:33:19
%S A224822 1,-2,0,-4,10,0,4,-16,0,-2,8,0,12,-8,0,-16,26,0,0,-24,0,-8,8,0,20,-10,
%T A224822 0,-4,32,0,8,-48,0,-8,16,0,10,-8,0,-32,40,0,8,-24,0,0,16,0,28,-18,0,
%U A224822 -24,40,0,4,-64,0,-8,8,0,32,-24,0,-16,58,0,16,-24,0,-16
%N A224822 Expansion of phi(-q) * phi(-q^3)^2 in powers of q where phi() is a Ramanujan theta function.
%C A224822 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A224822 G. C. Greubel, <a href="/A224822/b224822.txt">Table of n, a(n) for n = 0..2500</a>
%H A224822 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A224822 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A224822 Expansion of eta(q)^2 * eta(q^3)^4 / (eta(q^2) * eta(q^6)^2) in powers of q.
%F A224822 Euler transform of period 6 sequence [-2, -1, -6, -1, -2, -3, ...].
%F A224822 G.f.: (Sum_{k in Z} (-1)^k * x^k^2) * (Sum_{k in Z} (-1)^k * x^(3*k^2))^2.
%F A224822 a(3*n + 2) = 0. a(2*n) = A028967(n). a(3*n) = A224821(n).
%e A224822 G.f. = 1 - 2*q - 4*q^3 + 10*q^4 + 4*q^6 - 16*q^7 - 2*q^9 + 8*q^10 + 12*q^12 +
%e A224822 ...
%t A224822 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3]^2, {q, 0, n}];
%o A224822 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A)^4 / (eta(x^2 + A) * eta(x^6 + A)^2), n))};
%Y A224822 Cf. A028967, A224821.
%K A224822 sign
%O A224822 0,2
%A A224822 _Michael Somos_, Jul 20 2013