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

Author

Michael Somos, Jul 25 2015

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. = 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 + ...
		

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

Crossrefs

Cf. A213023, A213592, A213607, A213617.

Programs

Mathematica

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

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))};

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.