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A116604 Expansion of q^(-1/2) * eta(q)^3 * eta(q^4) * eta(q^12) / (eta(q^2)^2 * eta(q^3)) in powers of q.

%I A116604 #23 Feb 16 2025 08:33:00
%S A116604 1,-3,2,0,1,0,2,-6,2,0,0,0,3,-3,2,0,0,0,2,-6,2,0,2,0,1,-6,2,0,0,0,2,0,
%T A116604 4,0,0,0,2,-9,0,0,1,0,4,-6,2,0,0,0,2,0,2,0,0,0,2,-6,2,0,2,0,1,-6,4,0,
%U A116604 0,0,0,-6,2,0,0,0,4,-3,2,0,2,0,2,-6,0,0,0,0,3,0,2,0,0,0,2,-6,4,0,0,0,2,-12,2,0,0,0,4,0,0
%N A116604 Expansion of q^(-1/2) * eta(q)^3 * eta(q^4) * eta(q^12) / (eta(q^2)^2 * eta(q^3)) in powers of q.
%C A116604 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A116604 G. C. Greubel, <a href="/A116604/b116604.txt">Table of n, a(n) for n = 0..1000</a>
%H A116604 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A116604 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A116604 G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)) * (1 - x^k + x^(2*k)) * (1 + x^(6*k)).
%F A116604 G.f.: Sum_{k>=0} x^(3*k) / (1 + x^(6*k + 1)) - 2*x^(3*k + 1)  /(1 + x^(6*k+3)) + x^(3*k + 2) / (1 + x^(6*k + 5)).
%F A116604 Expansion of psi(q^2)^2 - 3 * q * psi(q^6)^2 in powers of q where psi() is a Ramanujan theta function.
%F A116604 Euler transform of period 12 sequence [ -3, -1, -2, -2, -3, 0, -3, -2, -2, -1, -3, -2, ...].
%F A116604 Moebius transform is period 24 sequence [ 1, -1, -4, 0, 1, 4, -1, 0, 4, -1, -1, 0, 1, 1, -4, 0, 1, -4, -1, 0, 4, 1, -1, 0, ...].
%F A116604 a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = -1 + 2 * (-1)^e, b(p^e) = e + 1 if p == 1, 5 (mod 12), b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).
%F A116604 G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 12 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A121450.
%F A116604 a(6*n + 3) = a(6*n + 5) = 0. a(6*n) = A002175(n). a(2*n) = A008441(n).
%e A116604 1 - 3*x + 2*x^2 + x^4 + 2*x^6 - 6*x^7 + 2*x^8 + 3*x^12 - 3*x^13 + ...
%e A116604 q - 3*q^3 + 2*q^5 + q^9 + 2*q^13 - 6*q^15 + 2*q^17 + 3*q^25 - 3*q^27 + ...
%t A116604 QP = QPochhammer; s = QP[q]^3*QP[q^4]*(QP[q^12]/(QP[q^2]^2*QP[q^3])) + O[q]^105; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 24 2015 *)
%o A116604 (PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -4, n/d) * [ -2, 1, 1][d%3 + 1]))}
%o A116604 (PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==2, 0, if( p==3, -1 + 2 * (-1)^e, if( p%12 < 6, e+1, (1 + (-1)^e) / 2)))))) }
%o A116604 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)), n))}
%Y A116604 Cf. A002175, A008441, A121450, A138741 (unsigned version).
%K A116604 sign
%O A116604 0,2
%A A116604 _Michael Somos_, Feb 18 2006, Apr 03 2008