A152243 Expansion of a(q) * f(-q)^4 where f() is a Ramanujan theta function and a() is a cubic AGM function.

1, 2, -22, 26, 25, -46, 26, -22, -45, 0, 74, 122, -46, -142, 0, -44, 2, 194, -214, 0, 121, 146, 52, -22, 0, -286, -118, -262, 315, 50, 314, 0, -382, 386, 0, -166, -92, 338, 26, 0, -286, -572, 0, 52, 0, 242, 122, 458, 289, 0, -44, -358, -142, 0, -550, 362, 482, -188, -502, 0, 315, -718, 698, -694

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 + 3 * (eta(q^3) * eta(q^9))^3 ) in powers of q^3.

a(n) = b(6*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, 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)) = 15552^(1/2) (t / i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A152244.

a(n) = A030208(3*n) = A152244(3*n).

Example

G.f. = 1 + 2*x - 22*x^2 + 26*x^3 + 25*x^4 - 46*x^5 + 26*x^6 - 22*x^7 + ...
G.f. = q + 2*q^7 - 22*q^13 + 26*q^19 + 25*q^25 - 46*q^31 + 26*q^37 - 22*q^43 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/6)*((eta[q^(1/3)]*eta[q^1])^3 + 3*(eta[q^1]*eta[q^3])^3), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 10 2018 *)

Prog

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

Crossrefs

Cf. A030208, A152244.

Sequence in context: A111751 A037416 A057871 * A074160 A368406 A137074

Adjacent sequences: A152240 A152241 A152242 * A152244 A152245 A152246

Keyword

sign

Author

Michael Somos, Nov 30 2008

Status

approved