A128636 Expansion of 3 * (b(q^2)^2 / b(q)) / (c(q)^2 / c(q^2)) in powers of q where b(), c() are cubic AGM theta functions.

1, 1, -3, 3, 5, -18, 15, 24, -75, 57, 86, -252, 183, 262, -744, 522, 725, -1998, 1365, 1852, -4986, 3336, 4436, -11736, 7719, 10103, -26322, 17067, 22040, -56682, 36306, 46336, -117867, 74700, 94378, -237744, 149277, 186926, -466836, 290706, 361126, -895014

Offset: 0

Author

Michael Somos, Mar 16 2007

Keywords

• sign

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. = 1 + q - 3*q^2 + 3*q^3 + 5*q^4 - 18*q^5 + 15*q^6 + 24*q^7 - 75*q^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A123633, A128641.

Programs

Mathematica

eta[x_] := x^(1/24)*QPochhammer[x]; A128636[n_] := SeriesCoefficient[(eta[q^6]/eta[q])*(eta[q^2]/eta[q^3])^5, {q, 0, n}]; Table[A128636[n], {n, 0, 50}] (* G. C. Greubel, Aug 21 2017 *)

PARI

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

Formula

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

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

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

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

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

A123633(n) = a(n) unless n = 0. Convolution inverse of A128641.

Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -(3/8)*sqrt(3) + (3/8)*sqrt(9 + 6*sqrt(3)). - Simon Plouffe, Mar 02 2021