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A261202 Expansion of phi(-x) * phi(-x^9) / f(-x^6)^2 in powers of x where phi(), f() are Ramanujan theta functions.

%I A261202 #10 Feb 16 2025 08:33:26
%S A261202 1,-2,0,0,2,0,2,-4,0,-4,8,0,5,-14,0,-8,20,0,14,-28,0,-20,44,0,28,-66,
%T A261202 0,-40,90,0,56,-124,0,-80,176,0,109,-244,0,-144,326,0,198,-432,0,-268,
%U A261202 580,0,349,-772,0,-456,1004,0,600,-1300,0,-780,1692,0,1001
%N A261202 Expansion of phi(-x) * phi(-x^9) / f(-x^6)^2 in powers of x where phi(), f() are Ramanujan theta functions.
%C A261202 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A261202 G. C. Greubel, <a href="/A261202/b261202.txt">Table of n, a(n) for n = 0..2500</a>
%H A261202 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261202 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261202 Expansion of q^(1/2) * eta(q)^2 * eta(q^9)^2 / (eta(q^2) * eta(q^6)^2 * eta(q^18)) in powers of q.
%F A261202 Euler transform of period 18 sequence [ -2, -1, -2, -1, -2, 1, -2, -1, -4, -1, -2, 1, -2, -1, -2, -1, -2, 0, ...].
%F A261202 a(3*n + 2) = 0.
%e A261202 G.f. = 1 - 2*x + 2*x^4 + 2*x^6 - 4*x^7 - 4*x^9 + 8*x^10 + 5*x^12 + ...
%e A261202 G.f. = 1/q - 2*q + 2*q^7 + 2*q^11 - 4*q^13 - 4*q^17 + 8*q^19 + ...
%t A261202 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^9] / QPochhammer[ x^6]^2, {x, 0, n}];
%o A261202 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^9 + A)^2 / (eta(x^2 + A) * eta(x^6 + A)^2 * eta(x^18 + A)), n))};
%K A261202 sign
%O A261202 0,2
%A A261202 _Michael Somos_, Aug 11 2015