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%I #32 Nov 14 2023 04:22:33
%S 1,2,4,2,2,0,4,4,4,2,0,0,2,4,8,0,2,0,4,4,0,4,0,0,4,2,8,2,4,0,0,4,4,0,
%T 0,0,2,4,8,4,0,0,8,4,0,0,0,0,2,6,4,0,4,0,4,0,8,4,0,0,0,4,8,4,2,0,0,4,
%U 0,0,0,0,4,4,8,2,4,0,8,4,0,2,0,0,4,0,8,0,0,0,0,8,0,4,0,0,4,4,12,0,2,0,0,4,8
%N Expansion of eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3) in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A123330/b123330.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>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of c(q)^2 / (3 * c(q^2)) in powers of q where c() is a cubic AGM theta function.
%F Expansion of phi(-x^3)^3 / phi(-x) where phi() is a Ramanujan theta function.
%F a(n) = 2*b(n) where b(n) is multiplicative and b(2^e) = (1 - 3*(-1)^e) / 2 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
%F Euler transform of period 6 sequence [ 2, 1, -4, 1, 2, -2, ...].
%F Moebius transform is period 6 sequence [ 2, 2, 0, -2, -2, 0, ...].
%F a(n) = 2 * A123331(n) if n>0. (-1)^n * a(n) = A113973(n).
%F G.f.: Product_{k>0} (1 + x^k)/(1 - x^k) * ((1 - x^(3*k)) / (1 + x^(3*k)))^3.
%F G.f.: 1 + 2 * Sum_{k>0} x^k / (1 - x^k + x^(2*k)) = theta_3(-x^3)^3 / theta_3(-x).
%F From _Michael Somos_, Aug 11 2009: (Start)
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u - v)^2 - 2 * u * w * (v - w).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (16/3)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A107760.
%F a(4*n) = a(3*n) = a(n). a(12*n + 10) = a(6*n + 5) = 0.
%F a(2*n + 1) = 2 * A033762(n). a(3*n + 1) = 2 * A033687(n). a(4*n + 1) = 2 * A112604(n). a(4*n + 3) = 2 * A112605(n). a(6*n + 1) = 2 * A097195(n). a(12*n + 1) = A123884(n). a(12*n + 7) = 4 * A121361(n). (End)
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(3*sqrt(3)) = 2.418399... (A275486). - _Amiram Eldar_, Nov 14 2023
%e G.f. = 1 + 2*q + 4*q^2 + 2*q^3 + 2*q^4 + 4*q^6 + 4*q^7 + 4*q^8 + 2*q^9 + ... - _Michael Somos_, Aug 11 2009
%t QP = QPochhammer; s = QP[q^2]*(QP[q^3]^6/(QP[q]^2*QP[q^6]^3)) + O[q]^105; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015 *)
%o (PARI) {a(n) = if( n<1, n==0, 2 * sumdiv(n, d, -(-1)^d * kronecker( -3, d)))}
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^6 / (eta(x + A)^2 * eta(x^6 + A)^3), n))}
%o (Sage) A = ModularForms( Gamma1(6), 1, prec=90).basis(); A[0] + 2*A[1] # _Michael Somos_, Sep 27 2013
%Y Cf. A033687, A033762, A097195, A107760, A112604, A112605, A113973, A121361, A123331, A123884, A275486.
%Y Cf. A000700, A000122, A010054, A121373.
%K nonn,easy
%O 0,2
%A _Michael Somos_, Sep 26 2006
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