A257651 Expansion of chi(x)^2 * f(-x^6)^3 in powers of x where chi(), f() are Ramanujan theta functions.

1, 2, 1, 2, 4, 4, 2, 0, 6, 6, 1, 4, 6, 8, 2, 0, 7, 6, 4, 6, 8, 8, 4, 0, 10, 8, 2, 6, 10, 12, 0, 0, 9, 14, 6, 6, 12, 8, 6, 0, 10, 12, 1, 10, 14, 8, 4, 0, 16, 14, 6, 8, 8, 16, 8, 0, 12, 14, 2, 10, 12, 16, 0, 0, 20, 10, 7, 8, 20, 20, 6, 0, 10, 16, 4, 10, 20, 12

Offset

0,2

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 = 0..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of phi(x^3) * f(x, x^5)^2 in powers of x where phi(), f() are Ramanujan theta functions.

Expansion of phi(x) * c(x^2) / 3 in powers of x where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.

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

Euler transform of period 12 sequence [ 2, -2, 2, 0, 2, -5, 2, 0, 2, -2, 2, -3, ...].

a(n) = a(4*n + 2) = A213592(2*n + 1). a(2*n) = A213592(n). a(2*n + 1) = 2 * A213607(n).

a(8*n + 2) = A213592(n). a(8*n + 3) = 2 * A213617(n). a(8*n + 5) = 4 * A213023(n). a(8*n + 6) = 2 * A213607(n). a(8*n + 7) = 0.

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/24) * exp(2*Pi/3) * Pi^(5/4) * 3^(1/2) * Gamma(11/12)^4 * Gamma(7/12)^3 * sqrt(2) * (3^(1/2)-1) / Gamma(2/3) / Gamma(3/4)^10 = A388884. - Simon Plouffe, Sep 21 2025

Example

G.f. = 1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 2*x^6 + 6*x^8 + 6*x^9 + ...
G.f. = q^2 + 2*q^5 + q^8 + 2*q^11 + 4*q^14 + 4*q^17 + 2*q^20 + 6*q^26 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^2 QPochhammer[ x^6]^3, {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)^2), n))};

Crossrefs

Cf. A213023, A213592, A213607, A213617.

Sequence in context: A263293 A230535 A349741 * A275122 A264569 A265601

Adjacent sequences: A257648 A257649 A257650 * A257652 A257653 A257654

Keyword

nonn

Author

Michael Somos, Jul 25 2015

Status

approved