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A122373 Expansion of (c(q)^3 + c(q^2)^3) / 27 in powers of q where c() is a cubic AGM theta function.

%I A122373 #34 Feb 16 2025 08:33:02
%S A122373 1,4,9,16,24,36,50,64,81,96,120,144,170,200,216,256,288,324,362,384,
%T A122373 450,480,528,576,601,680,729,800,840,864,962,1024,1080,1152,1200,1296,
%U A122373 1370,1448,1530,1536,1680,1800,1850,1920,1944,2112,2208,2304,2451,2404
%N A122373 Expansion of (c(q)^3 + c(q^2)^3) / 27 in powers of q where c() is a cubic AGM theta function.
%C A122373 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A122373 a(n) = n^2 and n > 0 if and only if n = 2^i * 3^j with i, j >=0 (numbers in A003586). - _Michael Somos_, Jun 08 2012
%H A122373 Vincenzo Librandi, <a href="/A122373/b122373.txt">Table of n, a(n) for n = 1..1000</a>
%H A122373 Kevin Acres and David Broadhurst, <a href="https://arxiv.org/abs/1810.07478">Eta quotients and Rademacher sums</a>, arXiv:1810.07478 [math.NT], 2018. See p. 3 eq. (3).
%H A122373 Mathew D. Rogers, <a href="https://arxiv.org/abs/0806.3590">Hypergeometric formulas for lattice sums and Mahler measures</a>, arXiv:0806.3590 [math.NT], 2008-2010. See p. 15, eq. (4.21).
%H A122373 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A122373 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A122373 Expansion of eta(q^2)^5 * eta(q^3)^4 * eta(q^6) / eta(q)^4 in powers of q.
%F A122373 a(n) is multiplicative with a(2^e) = 4^e, a(3^e) = 9^e, a(p^e) = (p^(2*e + 2) - f^(e+1)) / (p^2 - f) where f = 1 if p == 1 (mod 6), f = -1 if p == 5 (mod 6).
%F A122373 Euler transform of period 6 sequence [4, -1, 0, -1, 4, -6, ...].
%F A122373 G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 3^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is g.f. for A132000.
%F A122373 G.f.: Sum_{k>0} k^2 * x^k / (1 + x^k + x^(2*k)) * (1 + (1+(-1)^k)/8).
%F A122373 G.f.: Product_{k>0} (1 - x^k) * (1 + x^(3*k)) * (1 + x^k)^5 * (1 - x^(3*k))^5.
%F A122373 Expansion of psi(q)^2 * psi(q^3)^2 * phi(-q^3)^3 / phi(-q) in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Jun 23 2012
%F A122373 Expansion of c(q) * c(q^2) * b(q^2)^2 / (9 * b(q)) in powers of q where b(), c() are cubic AGM theta functions. - _Michael Somos_, Jun 23 2012
%F A122373 G.f.: Sum_{k>0} k^2 * x^k * (1 + x^(2*k)) / (1 + x^(2*k) + x^(4*k)). - _Michael Somos_, Jul 05 2020
%F A122373 Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Pi^3/(18*sqrt(3)) = 0.994526... (A346585). - _Amiram Eldar_, Dec 22 2023
%e A122373 G.f. = q + 4*q^2 + 9*q^3 + 16*q^4 + 24*q^5 + 36*q^6 + 50*q^7 + 64*q^8 + 81*q^9 + 96*q^10 + ...
%t A122373 terms = 50; QP = QPochhammer; s = QP[q^2]^5*QP[q^3]^4*(QP[q^6]/QP[q]^4) + O[q]^terms; CoefficientList[s, q] (* _Jean-François Alcover_, Jul 04 2017, from first formula *)
%o A122373 (PARI) {a(n) = my(A, p, e, f); if( n<0, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, p^(2*e), f =- (-1)^(p%3); (p^(2*e + 2) - f^(e+1)) / (p^2 - f))))};
%o A122373 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^4 * eta(x^6 + A) / eta(x + A)^4, n))};
%Y A122373 Cf. A003586, A132000, A346585.
%Y A122373 Cf. A000122, A000700, A010054, A121373.
%K A122373 nonn,easy,mult
%O A122373 1,2
%A A122373 _Michael Somos_, Aug 30 2006