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%I #16 Mar 12 2021 22:24:48
%S 1,-2,2,-2,0,0,1,-2,4,0,0,0,0,-4,2,-2,0,0,3,-2,2,-2,0,0,2,-2,2,0,0,0,
%T 0,-2,2,-2,0,0,3,-2,4,-2,0,0,0,-6,2,0,0,0,0,-2,4,0,0,0,2,-2,2,-4,0,0,
%U 1,0,2,0,0,0,0,-2,6,-2,0,0,2,-4,0,-2,0,0,4,-4
%N Expansion of f(x^2, -x^4) * f(-x, -x^5)^2 / f(-x^12, -x^12) in powers of x where f(, ) is Ramanujan's general theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A257921/b257921.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 psi(-x)^2 * psi(x^3)^2 / (psi(-x^2) * phi(-x^12)) in powers of x where phi(), psi() are Ramanujan theta functions.
%F Expansion of q^(-3/4) * eta(q)^2 * eta(q^4)^3 * eta(q^6)^4 * eta(q^24) / (eta(q^2)^3 * eta(q^3)^2 *eta(q^8) * eta(q^12)^2) in powers of q.
%F Euler transform of period 24 sequence [ -2, 1, 0, -2, -2, -1, -2, -1, 0, 1, -2, -2, -2, 1, 0, -1, -2, -1, -2, -2, 0, 1, -2, -2, ...].
%F a(n) = b(4*n + 3) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 11 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 7 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
%F a(n) = (-1)^n * A261119(n) = A134177(2*n + 1) = - A128580(2*n + 1) = - A115660(4*n+3).
%F a(6*n + 4) = a(6*n + 5) = 0.
%F a(2*n) = A257920(n). a(2*n + 1) = -2 * A259896(n). a(3*n) = A259668(n). a(6*n + 2) = 2 * A128591(n).
%e G.f. = 1 - 2*x + 2*x^2 - 2*x^3 + x^6 - 2*x^7 + 4*x^8 - 4*x^13 + 2*x^14 + ...
%e G.f. = q^3 - 2*q^7 + 2*q^11 - 2*q^15 + q^27 - 2*q^31 + 4*q^35 - 4*q^55 + ...
%t a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 3}, DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ -2, m/#] &]]];
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)])^2 / (2^(5/2) x^(3/4) EllipticTheta[ 2, Pi/4, x] EllipticTheta[ 4, 0, x^12]), {x, 0, n};
%o (PARI) {a(n) = if( n<0, 0, n = 4*n + 3; sumdiv(n, d, kronecker( 12, d) * kronecker( -2, n/d)))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^6 + A)^4 * eta(x^24 + A) / (eta(x^2 + A)^3 * eta(x^3 + A)^2 *eta(x^8 + A) * eta(x^12 + A)^2), n))};
%Y Cf. A115660, A128580, A128591, A134177, A257920, A259668, A259896, A261119.
%K sign
%O 0,2
%A _Michael Somos_, Jul 12 2015
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