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A246926 Expansion of phi(x)^2 * chi(x) * psi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

%I A246926 #18 Feb 16 2025 08:33:23
%S A246926 1,5,8,4,4,13,12,4,5,16,24,8,4,20,12,8,9,20,32,4,12,29,12,8,8,36,40,8,
%T A246926 8,20,24,16,8,25,40,12,12,32,24,12,13,48,40,8,8,40,36,8,16,20,56,16,
%U A246926 12,52,12,20,13,36,56,16,20,40,24,8,8,45,72,12,16,52
%N A246926 Expansion of phi(x)^2 * chi(x) * psi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
%C A246926 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A246926 G. C. Greubel, <a href="/A246926/b246926.txt">Table of n, a(n) for n = 0..2500</a>
%H A246926 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A246926 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A246926 Expansion of q^(-1/3) * eta(q^2)^12 * eta(q^3) * eta(q^12) / (eta(q)^5 * eta(q^4)^5 * eta(q^6)) in powers of q.
%F A246926 Euler transform of period 12 sequence [5, -7, 4, -2, 5, -7, 5, -2, 4, -7, 5, -3, ...].
%F A246926 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 72^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246927.
%F A246926 2 * a(n) = A246928(3*n + 1).
%e A246926 G.f. = 1 + 5*x + 8*x^2 + 4*x^3 + 4*x^4 + 13*x^5 + 12*x^6 + 4*x^7 + 5*x^8 + ...
%e A246926 G.f. = q + 5*q^4 + 8*q^7 + 4*q^10 + 4*q^13 + 13*q^16 + 12*q^19 + 4*q^22 + ...
%t A246926 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 3, 0, x]^2 EllipticTheta[ 2, Pi/4, x^(3/2)] / (2^(1/2) x^(3/8)), {x, 0, n}]; (* _Michael Somos_, Jan 08 2015 *)
%o A246926 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^12 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^5 * eta(x^4 + A)^5 * eta(x^6 + A)), n))};
%o A246926 (Magma) A := Basis( ModularForms( Gamma0(36), 3/2), 210); A[2] + 5*A[5];
%Y A246926 Cf. A246927, A246928.
%K A246926 nonn
%O A246926 0,2
%A A246926 _Michael Somos_, Sep 07 2014