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A230534 Expansion of phi(x) / phi(x^2) * f(-x, -x^7) / f(-x^3, -x^5) in powers of x where phi(), f() are Ramanujan theta functions.

%I A230534 #22 Feb 16 2025 08:33:20
%S A230534 1,1,-4,-1,11,-1,-20,2,34,1,-64,-2,113,-2,-180,2,284,4,-452,-4,698,-5,
%T A230534 -1044,6,1545,6,-2276,-7,3308,-9,-4724,10,6680,13,-9408,-15,13130,-17,
%U A230534 -18112,20,24821,21,-33864,-25,45882,-28,-61696,32,82514,39,-109892,-43
%N A230534 Expansion of phi(x) / phi(x^2) * f(-x, -x^7) / f(-x^3, -x^5) in powers of x where phi(), f() are Ramanujan theta functions.
%C A230534 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A230534 Robert G. Wilson v, <a href="/A230534/b230534.txt">Table of n, a(n) for n = 0..10000</a> (first 2501 terms from G. C. Greubel)
%H A230534 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A230534 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A230534 Expansion of (phi(x) / psi(x^2)^2) * (phi(x) - phi(x^2)) / (2*x) in powers of x where phi(), psi() are Ramanujan theta functions.
%F A230534 Expansion of q^(-1) * phi(q^2) * (phi(q^2) - phi(q^4)) / (phi(q^4) * (phi(q) - phi(q^4))) in powers of q^2 where phi() is a Ramanujan theta function.
%F A230534 Euler transform of period 8 sequence [1, -5, 3, 2, 3, -5, 1, 0, ...].
%F A230534 a(2*n + 1) = A226559(n).
%F A230534 Convolution of A080015 and A092869.
%e A230534 G.f. = 1 + x - 4*x^2 - x^3 + 11*x^4 - x^5 - 20*x^6 + 2*x^7 + 34*x^8 + ...
%e A230534 G.f. = q + q^3 - 4*q^5 - q^7 + 11*q^9 - q^11 - 20*q^13 + 2*q^15 + ...
%t A230534 a[ n_] := SeriesCoefficient[ 2 q^(-1/2) EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q]^2 (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2]), {q, 0, n}];
%o A230534 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^6) * sum( k=1, sqrtint(n+1), x^k^2 - x^(2*k^2), x * A) / x, n))};
%Y A230534 Cf. A080015, A092869, A226559.
%K A230534 sign
%O A230534 0,3
%A A230534 _Michael Somos_, Oct 22 2013