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A213607 Expansion of psi(x^4) * f(-x^3)^3 / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

%I A213607 #15 Feb 16 2025 08:33:17
%S A213607 1,1,2,0,3,2,4,0,3,3,4,0,4,3,6,0,7,3,4,0,6,5,4,0,7,4,8,0,7,5,8,0,5,4,
%T A213607 10,0,8,5,6,0,7,7,8,0,11,5,10,0,9,8,8,0,5,4,12,0,14,5,8,0,10,8,8,0,11,
%U A213607 8,10,0,10,9,14,0,10,5,10,0,15,7,6,0,10,9
%N A213607 Expansion of psi(x^4) * f(-x^3)^3 / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.
%C A213607 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A213607 G. C. Greubel, <a href="/A213607/b213607.txt">Table of n, a(n) for n = 0..1000</a>
%H A213607 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A213607 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A213607 Expansion of q^(-5/6) * eta(q^3)^3 * eta(q^8)^2 / (eta(q) * eta(q^4)) in powers of q.
%F A213607 Euler transform of period 24 sequence [ 1, 1, -2, 2, 1, -2, 1, 0, -2, 1, 1, -1, 1, 1, -2, 0, 1, -2, 1, 2, -2, 1, 1, -3, ...].
%F A213607 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 32^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A213618.
%F A213607 a(4*n + 3) = 0.  a(4*n + 2) = 2 * A213023(n).
%e A213607 1 + x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^6 + 3*x^8 + 3*x^9 + 4*x^10 + ...
%e A213607 q^5 + q^11 + 2*q^17 + 3*q^29 + 2*q^35 + 4*q^41 + 3*q^53 + 3*q^59 + 4*q^65 + ...
%t A213607 QP := QPochhammer; a[n_]:= SeriesCoefficient[(QP[q^3]^3*QP[q^8]^2 )/( QP[q]*QP[q^4]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Jan 07 2018 *)
%o A213607 (PARI) {a(n) = local(A); if ( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^8 + A)^2 / (eta(x + A) * eta(x^4 + A)), n))}
%Y A213607 Cf. A213023, A213618.
%K A213607 nonn
%O A213607 0,3
%A A213607 _Michael Somos_, Jun 16 2012