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

1,2

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).

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

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

Example

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

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 *)

Prog

(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))};

Crossrefs

Cf. A128632, A128639.

Sequence in context: A135266 A124123 A189714 * A036630 A338229 A102841

Adjacent sequences: A128635 A128636 A128637 * A128639 A128640 A128641

Keyword

nonn,changed

Author

Michael Somos, Mar 16 2007

Extensions

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

Status

approved