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A260415 Expansion of f(x, x^2) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.

%I A260415 #21 Feb 16 2025 08:33:26
%S A260415 1,1,1,0,1,2,1,1,1,2,1,1,1,1,0,2,1,0,0,1,2,1,2,1,0,1,2,1,1,1,3,0,1,1,
%T A260415 1,3,0,0,0,1,2,0,1,2,1,0,1,0,2,1,2,1,0,1,1,3,0,1,0,1,3,2,1,2,0,2,0,1,
%U A260415 1,0,2,1,1,0,2,1,0,2,1,1,0,1,1,1,0,2,1
%N A260415 Expansion of f(x, x^2) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.
%C A260415 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260415 G. C. Greubel, <a href="/A260415/b260415.txt">Table of n, a(n) for n = 0..2500</a>
%H A260415 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260415 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260415 Expansion of q^(-5/24) * eta(q^2) * eta(q^3)^2 * eta(q^8) * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^24)) in powers of q.
%F A260415 Euler transform of period 24 sequence [ 1, 0, -1, 1, 1, -1, 1, 0, -1, 0, 1, -2, 1, 0, -1, 0, 1, -1, 1, 1, -1, 0, 1, -2, ...].
%e A260415 G.f. = 1 + x + x^2 + x^4 + 2*x^5 + x^6 + x^7 + x^8 + 2*x^9 + x^10 + x^11 + ...
%e A260415 G.f. = q^5 + q^29 + q^53 + q^101 + 2*q^125 + q^149 + q^173 + q^197 + ...
%t A260415 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x] QPochhammer[ x^3]^2 QPochhammer[ -x^6, x^12] / (2^(1/2) x^(1/4) QPochhammer[ x]), {x, 0, n}];
%t A260415 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^12] / (QPochhammer[ x, x^2] QPochhammer[ x^4, x^8]), {x, 0, n}];
%o A260415 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A)), n))};
%K A260415 sign
%O A260415 0,6
%A A260415 _Michael Somos_, Jul 27 2015