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

0,3

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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).

Example

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 + ...

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}];

Prog

(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))};

Crossrefs

Cf. A181648, A185220.

Sequence in context: A364098 A363521 A166050 * A221185 A242746 A363258

Adjacent sequences: A259652 A259653 A259654 * A259656 A259657 A259658

Keyword

nonn

Author

Michael Somos, Jul 02 2015

Status

approved