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A246836 Expansion of phi(x) * psi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A246836 #13 Feb 16 2025 08:33:23
%S A246836 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 A246836 -6,-12,8,6,-10,-12,4,14,-8,-4,16,10,-8,-8,9,10,-12,-12,8,12,-12,-4,
%U A246836 20,10,-6,-20,8,6,-10,-12,8,20,-18,-8,11,12,-12,-16,8,6,-20
%N A246836 Expansion of phi(x) * psi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A246836 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A246836 G. C. Greubel, <a href="/A246836/b246836.txt">Table of n, a(n) for n = 0..1000</a>
%H A246836 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A246836 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A246836 Expansion of q^(-1/2) * eta(q^2)^7 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)^4) in powers of q.
%F A246836 Euler transform of period 8 sequence [ 2, -5, 2, -1, 2, -5, 2, -3, ...].
%F A246836 G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 32 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246835.
%F A246836 a(n) = (-1)^floor(n/2) * A045828(n). a(n) = (-1)^n * A246815(n).
%F A246836 a(2*n) = A246835(n). a(2*n + 1) = 2 * A246833(n).
%e A246836 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 A246836 G.f. = q + 2*q^3 - 2*q^5 - 4*q^7 + 3*q^9 + 2*q^11 - 6*q^13 - 4*q^15 + 4*q^17 + ...
%t A246836 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x]^2 EllipticTheta[ 3, 0, x] / (2 x^(1/2)), {x, 0, n}];
%o A246836 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^4), n))};
%Y A246836 Cf. A045828, A246833, A246835.
%K A246836 sign
%O A246836 0,2
%A A246836 _Michael Somos_, Sep 04 2014