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%I #24 Dec 28 2023 01:52:57
%S 1,3,6,8,9,12,14,18,18,20,24,24,31,27,30,32,36,48,38,42,42,44,54,48,
%T 57,54,54,72,60,60,62,72,84,68,72,72,74,93,96,80,81,84,108,90,90,112,
%U 96,120,98,108,102,104,144,108,110,114,114,144,126,144,133,126,156,128
%N Expansion of q^(-1) * c(q^2) * (c(q) - c(q^4)) / 9 in powers of q^2 where c() is a cubic AGM 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="/A185717/b185717.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 phi(-x^3)^4 / (chi(-x) * chi(-x^3))^3 in powers of x where phi(), chi() are Ramanujan theta functions.
%F Euler transform of period 6 sequence [ 3, 0, -2, 0, 3, -4, ...].
%F a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 3^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A118271.
%F a(3*n + 1) = 3 * a(n). A078708(2*n + 1) = A121443(2*n + 1) = A124449(2*n + 1).
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/9 = 1.0966227... (A100044). - _Amiram Eldar_, Dec 28 2023
%e 1 + 3*x + 6*x^2 + 8*x^3 + 9*x^4 + 12*x^5 + 14*x^6 + 18*x^7 + 18*x^8 + ...
%e q + 3*q^3 + 6*q^5 + 8*q^7 + 9*q^9 + 12*q^11 + 14*q^13 + 18*q^15 + ...
%t A185717[n_] := SeriesCoefficient[(QPochhammer[q^3, q^3]/QPochhammer[-q^3, q^3])^4*(1/(QPochhammer[q, q^2]*QPochhammer[q^3, q^6])^3), {q, 0, n}];
%t Table[A185717[n], {n, 0, 50}] (* _G. C. Greubel_, Jul 10 2017 *)
%o (PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, if( (n/d) % 3, 1, 0) * d))}
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^5 / (eta(x + A)^3 * eta(x^6 + A)), n))}
%Y Cf. A078708, A100044, A118271, A121443, A124449.
%Y Cf. A000122, A000700, A010054, A121373.
%K nonn,easy
%O 0,2
%A _Michael Somos_, Feb 10 2011
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