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

1, 4, 9, 16, 24, 36, 50, 64, 81, 96, 120, 144, 170, 200, 216, 256, 288, 324, 362, 384, 450, 480, 528, 576, 601, 680, 729, 800, 840, 864, 962, 1024, 1080, 1152, 1200, 1296, 1370, 1448, 1530, 1536, 1680, 1800, 1850, 1920, 1944, 2112, 2208, 2304, 2451, 2404

Offset

1,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Kevin Acres and David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See p. 3 eq. (3).

Mathew D. Rogers, Hypergeometric formulas for lattice sums and Mahler measures, arXiv:0806.3590 [math.NT], 2008-2010. See p. 15, eq. (4.21).

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

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

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).

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

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.

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

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

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

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

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

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

Example

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 + ...

Mathematica

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 *)

Prog

(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))))};
(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))};

Crossrefs

Cf. A003586, A132000, A346585.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A052117 A033611 A033615 * A070458 A070457 A070456

Adjacent sequences: A122370 A122371 A122372 * A122374 A122375 A122376

Keyword

nonn,easy,mult

Author

Michael Somos, Aug 30 2006

Status

approved