%I #33 Dec 22 2023 08:41:35

%S 1,4,9,16,24,36,50,64,81,96,120,144,170,200,216,256,288,324,362,384,

%T 450,480,528,576,601,680,729,800,840,864,962,1024,1080,1152,1200,1296,

%U 1370,1448,1530,1536,1680,1800,1850,1920,1944,2112,2208,2304,2451,2404

%N Expansion of (c(q)^3 + c(q^2)^3) / 27 in powers of q 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).

%C 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 Vincenzo Librandi, <a href="/A122373/b122373.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 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 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 eta(q^2)^5 * eta(q^3)^4 * eta(q^6) / eta(q)^4 in powers of q.

%F 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 Euler transform of period 6 sequence [4, -1, 0, -1, 4, -6, ...].

%F 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 G.f.: Sum_{k>0} k^2 * x^k / (1 + x^k + x^(2*k)) * (1 + (1+(-1)^k)/8).

%F G.f.: Product_{k>0} (1 - x^k) * (1 + x^(3*k)) * (1 + x^k)^5 * (1 - x^(3*k))^5.

%F 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 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 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 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 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 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 (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 (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 Cf. A003586, A132000, A346585.

%Y Cf. A000122, A000700, A010054, A121373.

%K nonn,easy,mult

%O 1,2

%A _Michael Somos_, Aug 30 2006