A152244 Expansion of a(x) * f(-x^3)^4 in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM function.

1, 6, 0, 2, -18, 0, -22, 0, 0, 26, 12, 0, 25, 54, 0, -46, 0, 0, 26, -132, 0, -22, 0, 0, -45, 0, 0, 0, 156, 0, 74, -36, 0, 122, 0, 0, -46, 150, 0, -142, -162, 0, 0, 0, 0, -44, -276, 0, 2, 0, 0, 194, 0, 0, -214, 156, 0, 0, 396, 0, 121, 0, 0, 146, -132, 0, 52, 0, 0, -22, 0, 0, 0, -270, 0, -286, 0, 0, -118

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/2) *( (eta(q)*eta(q^3))^3 + 9*(eta(q^3)*eta(q^9))^3 ) in powers of q.

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = -2 * (-3)^e if e>0, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) if p == 1 (mod 6).

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 139968^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A152243.

a(3*n) = A152243(n). a(3*n + 1) = 6 * A030208(n). a(3*n + 2) = 0.

Example

G.f. = 1 + 6*x + 2*x^3 - 18*x^4 - 22*x^6 + 26*x^9 + 12*x^10 + 25*x^11 + ...
G.f. = q + 6*q^3 + 2*q^7 - 18*q^9 - 22*q^13 + 26*q^19 + 12*q^21 + 25*q^25 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^3])^3 + 9 x (QPochhammer[ x^3] QPochhammer[ x^9])^3 , {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^3 + 9 * x * (eta(x^3 + A) * eta(x^9 + A))^3, n))};

Crossrefs

Cf. A030208, A152243.

Sequence in context: A201331 A366349 A075092 * A283634 A179641 A110993

Adjacent sequences: A152241 A152242 A152243 * A152245 A152246 A152247

Keyword

sign

Author

Michael Somos, Nov 30 2008

Status

approved