A259655 Expansion of psi(x^2) * f(-x^3)^3 / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

1, 1, 3, 1, 4, 1, 5, 2, 5, 2, 5, 3, 7, 1, 7, 2, 9, 3, 7, 2, 6, 4, 11, 3, 8, 3, 10, 3, 8, 4, 9, 3, 14, 2, 10, 2, 15, 6, 7, 5, 7, 3, 14, 5, 14, 3, 16, 5, 8, 4, 13, 5, 13, 3, 12, 4, 18, 5, 14, 4, 13, 5, 15, 4, 15, 5, 16, 7, 9, 6, 11, 7, 22, 3, 16, 3, 19, 7, 16

Offset: 0

Author

Michael Somos, Jul 02 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 + x + 3*x^2 + x^3 + 4*x^4 + x^5 + 5*x^6 + 2*x^7 + 5*x^8 + ...
G.f. = q^7 + q^19 + 3*q^31 + q^43 + 4*q^55 + q^67 + 5*q^79 + 2*q^91 + ...
		

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. A181648, A185220.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^2, x^2]^2 QPochhammer[ x^3]^3, {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x]), {x, 0, n}];

PARI

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

Formula

Expansion of psi(x^2) * c(x) / (3 * x^(1/3)) in powers of x where psi() is a Ramanujan theta function and c() is a cubic AGM function.

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

Expansion of q^(-7/12) * eta(q^3)^3 * eta(q^4)^2 / (eta(q) * eta(q^2)) in powers of q.

Euler transform of period 12 sequence [ 1, 2, -2, 0, 1, -1, 1, 0, -2, 2, 1, -3, ...].

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

a(2*n) = A185220(n). a(2*n + 1) = A181648(n).