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%I #20 Mar 12 2021 22:24:44
%S 1,2,1,0,0,2,2,0,2,2,1,0,0,2,0,0,3,2,0,0,0,4,2,0,2,0,2,0,0,2,0,0,1,2,
%T 2,0,0,2,2,0,2,4,1,0,0,2,0,0,2,2,0,0,0,0,2,0,4,2,0,0,0,4,0,0,2,2,3,0,
%U 0,0,2,0,2,4,0,0,0,2,0,0,1,2,0,0,0,2,4,0,0,2,2,0,0,2,0,0,4,2,2,0,0,4,0,0,2
%N Expansion of f(x, x^5)^2 in powers of x where f(, ) is Ramanujan's general theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A122856/b122856.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of (chi(x) * psi(-x^3))^2 in powers of x where chi(), psi() are Ramanujan theta functions.
%F Expansion of q^(-2/3) * (eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)))^2 in powers of q.
%F Euler transform of period 12 sequence [2, -2, 0, 0, 2, -2, 2, 0, 0, -2, 2, -2, ...].
%F a(4*n + 3) = a(8*n + 4) = 0. a(n) = A002654(3*n + 2) = A035154(3*n + 2) = A113446(3*n + 2). a(2*n) = A122865(n). a(4*n + 1) = 2 * A121444(n). a(4*n + 2) = A122856(n).
%F a(n) = (-1)^n * A258278(n). Convolution square of A089801.
%e G.f. = 1 + 2*x + x^2 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + 2*x^13 + ...
%e G.f. = q^2 + 2*q^5 + q^8 + 2*q^17 + 2*q^20 + 2*q^26 + 2*q^29 + q^32 + ...
%t a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 2}, Sum[ KroneckerSymbol[ -4, d], {d, Divisors@m}]]]; (* _Michael Somos_, Nov 14 2011 *)
%t QP = QPochhammer; s = (QP[q^2]^2*QP[q^3]*(QP[q^12]/(QP[q]*QP[q^4]*QP[q^6]) ))^2 + O[q]^105; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 30 2015, adapted from PARI *)
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x^(1/3)] - EllipticTheta[ 3, 0, x^3])^2 / (4 x^(2/3)), {x, 0, n}]; (* _Michael Somos_, Jan 19 2017 *)
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)])^2 / (2 x^(3/4)), {x, 0, n}]; (* _Michael Somos_, Jan 19 2017 *)
%o (PARI) {a(n) = if( n<0, 0, n = 3*n+2; sumdiv(n, d, (d%4==1) - (d%4==3)))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)))^2, n))};
%Y Cf. A002654, A035154, A089801, A113446, A121444, A122865, A258278.
%K nonn
%O 0,2
%A _Michael Somos_, Sep 14 2006
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