A164616 Expansion of c(-q) * c(-q^3) / c(q^2)^2 in powers of q where c() is a cubic AGM theta function.

1, -1, 0, 1, -2, 0, 4, -5, 0, 10, -12, 0, 20, -26, 0, 39, -50, 0, 76, -92, 0, 140, -168, 0, 244, -295, 0, 415, -496, 0, 696, -818, 0, 1140, -1332, 0, 1820, -2126, 0, 2861, -3324, 0, 4448, -5126, 0, 6816, -7824, 0, 10292, -11793, 0, 15372, -17548, 0, 22756

Offset

0,5

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

Expansion of eta(q) * eta(q^4) * eta(q^18)^9 / (eta(q^2) * eta(q^3)^2 * eta(q^9)^3 * eta(q^12)^2 * eta(q^36)^3) 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 A258108. - Michael Somos, May 20 2015

a(3*n + 2) = 0. a(3*n) = A164617(n). a(3*n + 1) = -A132977(n).

Convolution inverse of A164615.

a(n) = (-1)^n * A258100(n). - Michael Somos, May 20 2015

Example

G.f. = 1 - q + q^3 - 2*q^4 + 4*q^6 - 5*q^7 + 10*q^9 - 12*q^10 + 20*q^12 + ...

Mathematica

eta[x_] := QPochhammer[x]; A164616[n_] := SeriesCoefficient[eta[q]* eta[q^4]*eta[q^18]^9/(eta[q^2]*eta[q^3]^2*eta[q^9]^3*eta[q^12]^2* eta[q^36]^3), {q, 0, n}]; Table[A164616[n], {n, 0, 50}] (* G. C. Greubel, Aug 10 2017 *)

Prog

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

Crossrefs

Cf. A132977, A164615, A164617, A258100, A258108.

Sequence in context: A162397 A091684 A100050 * A258100 A173335 A335118

Adjacent sequences: A164613 A164614 A164615 * A164617 A164618 A164619

Keyword

sign

Author

Michael Somos, Aug 17 2009

Status

approved