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
Examples
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 + ...
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.
Programs
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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 *)
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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))};
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
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