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A246954 Expansion of phi(-x) * psi(-x)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A246954 #12 Feb 16 2025 08:33:23
%S A246954 1,-4,5,-4,8,-8,5,-12,8,-4,16,-12,9,-12,8,-12,16,-16,8,-16,17,-8,24,
%T A246954 -8,8,-28,16,-12,16,-20,13,-24,24,-8,16,-16,16,-28,24,-12,32,-16,13,
%U A246954 -28,8,-20,32,-32,8,-20,24,-16,40,-16,16,-32,25,-20,24,-24,24,-28
%N A246954 Expansion of phi(-x) * psi(-x)^2 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A246954 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A246954 G. C. Greubel, <a href="/A246954/b246954.txt">Table of n, a(n) for n = 0..1000</a>
%H A246954 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A246954 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A246954 Expansion of phi(-x)^2 * psi(x^2) = psi(-x)^4 / psi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
%F A246954 Expansion of q^(-1/4) * eta(q)^4 * eta(q^4)^2 / eta(q^2)^3 in powers of q.
%F A246954 Euler transform of period 4 sequence [ -4, -1, -4, -3, ...].
%F A246954 G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 16 * (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A246953.
%F A246954 G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^(2*k))^2 / (1 + x^k).
%F A246954 a(n) = (-1)^n * A045834(n). a(2*n) = A213022(n). a(2*n + 1) = - 4 * A045831(n).
%e A246954 G.f. = 1 - 4*x + 5*x^2 - 4*x^3 + 8*x^4 - 8*x^5 + 5*x^6 - 12*x^7 + 8*x^8 + ...
%e A246954 G.f. = q - 4*q^5 + 5*q^9 - 4*q^13 + 8*q^17 - 8*q^21 + 5*q^25 - 12*q^29 + ...
%t A246954 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x]^2 EllipticTheta[ 2, 0, x]/(2 x^(1/4)), {x, 0, n}]; Table[a[n], {n, 0, 60}]
%o A246954 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A)^2 / eta(x^2 + A)^3, n))};
%Y A246954 Cf. A045831, A045834, A246953.
%K A246954 sign
%O A246954 0,2
%A A246954 _Michael Somos_, Sep 08 2014