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A255257 Expansion of psi(x) * phi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A255257 #10 Feb 16 2025 08:33:25
%S A255257 1,1,-4,-3,4,0,1,4,0,4,-3,-4,-4,-8,8,1,-4,0,0,4,0,5,4,8,-4,-4,4,-8,-3,
%T A255257 -4,4,-4,0,0,-8,4,1,0,-8,0,4,8,8,8,0,1,0,-8,8,-4,-4,-8,12,4,-12,1,-4,
%U A255257 0,0,-4,-8,4,-8,0,0,-8,1,12,8,8,0,-8,8,0,8,4,0
%N A255257 Expansion of psi(x) * phi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A255257 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A255257 G. C. Greubel, <a href="/A255257/b255257.txt">Table of n, a(n) for n = 0..1000</a>
%H A255257 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A255257 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A255257 Expansion of q^(-1/8) * eta(q^2)^6 / (eta(q) * eta(q4)^2) in powers of q.
%F A255257 Euler transform of period 4 sequence [ 1, -5, 1, -3, ...].
%F A255257 a(n) = A034950(4*n).
%e A255257 G.f. = 1 + x - 4*x^2 - 3*x^3 + 4*x^4 + x^6 + 4*x^7 + 4*x^9 - 3*x^10 + ...
%e A255257 G.f. = q + q^9 - 4*q^17 - 3*q^25 + 4*q^33 + q^49 + 4*q^57 + 4*q^73 + ...
%t A255257 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] ^2 EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}];
%o A255257 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 / (eta(x + A) * eta(x^4 + A)^2), n))};
%Y A255257 Cf. A034950.
%K A255257 sign
%O A255257 0,3
%A A255257 _Michael Somos_, Feb 19 2015