A128638 Expansion of q * (psi(q^3)^3 / psi(q)) / (phi(-q)^3 / phi(-q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.

1, 5, 19, 61, 174, 455, 1112, 2573, 5689, 12102, 24900, 49759, 96902, 184408, 343722, 628717, 1130418, 2000669, 3489788, 6005910, 10207688, 17147892, 28494120, 46865519, 76342903, 123236446, 197233723, 313106264, 493231830, 771301986, 1197743552, 1847606573

Offset: 1

Author

Michael Somos, Mar 16 2007

Keywords

• nonn

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = q + 5*q^2 + 19*q^3 + 61*q^4 + 174*q^5 + 455*q^6 + 1112*q^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A128632, A128639.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^6] / QPochhammer[ q])^5 (QPochhammer[ q^2] / QPochhammer[ q^3]), {q, 0, n}]; (* Michael Somos, Jun 08 2015 *)
nmax = 40; Rest[CoefficientList[Series[x * Product[((1 - x^(6*k)) / (1-x^k))^5 * ((1 - x^(2*k)) / (1 - x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 08 2015 *)

PARI

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^6 + A) / eta(x + A))^5 * eta(x^2 + A) / eta(x^3 + A), n))};

Formula

Expansion of (1/3) * (c(q^2)^2 / c(q)) / (b(q)^2 / b(q^2)) in powers of q where b(), c() are cubic AGM theta functions.

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

Euler transform of period 6 sequence [ 5, 4, 6, 4, 5, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1 + 8*u) * (1 + 9*v) - (u-v)^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1/72) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128632.

G.f.: x * Product_{k>0} ((1 - x^(6*k)) / (1-x^k))^5 * ((1 - x^(2*k)) / (1 - x^(3*k))).

8 * a(n) = A128639(n) unless n = 0. Convolution inverse of A128632.

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (72 * 2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

Extensions

Edited by N. J. A. Sloane, Apr 01 2008