A164615 Expansion of c(q^2)^2 / (c(-q) * c(-q^3)) in powers of q where c() is a cubic AGM theta function.

1, 1, 1, 0, 1, 2, 0, 0, 1, 0, -2, -4, 0, -2, -8, 0, 1, -2, 0, 4, 14, 0, 4, 24, 0, -1, 6, 0, -8, -38, 0, -8, -63, 0, 2, -16, 0, 14, 92, 0, 14, 150, 0, -4, 36, 0, -24, -208, 0, -23, -329, 0, 6, -78, 0, 40, 440, 0, 38, 684, 0, -10, 160, 0, -63, -884, 0, -60

Offset: 0

Author

Michael Somos, Aug 17 2009

Keywords

• sign

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 + q + q^2 + q^4 + 2*q^5 + q^8 - 2*q^10 - 4*q^11 - 2*q^13 - 8*q^14 + ...
		

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. A128111, A164614, A164616, A182034. A258111.

Programs

Mathematica

eta[x_] := QPochhammer[x]; A164615[n_] := SeriesCoefficient[eta[q^2]* eta[q^3]^2*eta[q^9]^3*eta[q^12]^2*eta[q^36]^3/(eta[q]*eta[q^4] *eta[q^18]^9), {q, 0, n}]; Table[A164615[n], {n,0,50}] (* G. C. Greubel, Aug 10 2017 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^9 + A)^3 * eta(x^12 + A)^2 * eta(x^36 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^9), n))};

Formula

Expansion of (chi(q^3) * psi(-q^3)^2)^2 / (psi(-q) * f(q^9)^2) in powers of q where psi(), chi(), f() are Ramanujan theta functions.

Expansion of eta(q^2) * eta(q^3)^2 * eta(q^9)^3 * eta(q^12)^2 * eta(q^36)^3 / (eta(q) * eta(q^4) * eta(q^18)^9) in powers of q.

Euler transform of period 36 sequence [ 1, 0, -1, 1, 1, -2, 1, 1, -4, 0, 1, -3, 1, 0, -1, 1, 1, 4, 1, 1, -1, 0, 1, -3, 1, 0, -4, 1, 1, -2, 1, 1, -1, 0, 1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258111. - Michael Somos, May 20 2015

a(3*n) = 0 unless n=0. a(3*n + 1) = A128111(n). a(3*n + 2) = A164614(n).

Convolution inverse of A164616.

a(n) = (-1)^n * A182034(n). - Michael Somos, May 20 2015