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A246815 Expansion of phi(-x) * psi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A246815 #10 Feb 16 2025 08:33:23
%S A246815 1,-2,-2,4,3,-2,-6,4,4,-6,-4,4,7,-8,-2,8,8,-4,-10,4,4,-10,-10,8,9,-4,
%T A246815 -6,12,8,-6,-10,12,4,-14,-8,4,16,-10,-8,8,9,-10,-12,12,8,-12,-12,4,20,
%U A246815 -10,-6,20,8,-6,-10,12,8,-20,-18,8,11,-12,-12,16,8,-6,-20
%N A246815 Expansion of phi(-x) * psi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A246815 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A246815 G. C. Greubel, <a href="/A246815/b246815.txt">Table of n, a(n) for n = 0..1000</a>
%H A246815 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A246815 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A246815 Expansion of q^(-1/2) * eta(q)^2 * eta(q^2) * eta(q^8)^2 / eta(q^4)^2 in powers of q
%F A246815 Euler transform of period 8 sequence [ -2, -3, -2, -1, -2, -3, -2, -3, ...].
%F A246815 a(n) = (-1)^floor((n+1) / 2) * A045828(n).
%F A246815 a(4*n) = A213622(n). a(4*n + 3) = 4 * A033763(n).
%e A246815 G.f. = 1 - 2*x - 2*x^2 + 4*x^3 + 3*x^4 - 2*x^5 - 6*x^6 + 4*x^7 + 4*x^8 + ...
%e A246815 G.f. = q - 2*q^3 - 2*q^5 + 4*q^7 + 3*q^9 - 2*q^11 - 6*q^13 + 4*q^15 + ...
%t A246815 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, Pi/4, x]^2 / (2 x^(1/2)), {x, 0, n}];
%o A246815 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A) * eta(x^8 + A)^2 / eta(x^4 + A)^2, n))};
%Y A246815 Cf. A033763, A045828, A213622.
%K A246815 sign
%O A246815 0,2
%A A246815 _Michael Somos_, Sep 03 2014