A134005 Expansion of (chi(-x) * chi(-x^19))^2 in powers of x where chi() is a Ramanujan theta function.

1, -2, 1, -2, 4, -4, 5, -6, 9, -12, 13, -16, 21, -26, 29, -36, 46, -54, 62, -76, 94, -108, 126, -150, 179, -210, 239, -282, 335, -384, 440, -512, 597, -684, 781, -902, 1041, -1186, 1347, -1544, 1768, -2006, 2268, -2584, 2941, -3318, 3742, -4236, 4792, -5392

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(5/3) * (eta(q) * eta(q^19) / (eta(q^2) * eta(q^38)))^2 in powers of q.

Euler transform of period 38 sequence [ -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -4, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (342 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134004.

G.f.: (Product_{k>0} (1 + x^k) * (1 + x^(19*k)))^-2.

a(2*n + 1) = -2 * (A112199(n) + A134004(n-2)). Convolution inverse of A134004.

a(n) ~ (-1)^n * 5^(1/4) * exp(2*Pi*sqrt(5*n/57)) / (2 * 57^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Example

G.f. = 1 - 2*x + x^2 - 2*x^3 + 4*x^4 - 4*x^5 + 5*x^6 - 6*x^7 + 9*x^8 - 12*x^9 + ...
G.f. = q^-5 - 2*q^-2 + q - 2*q^4 + 4*q^7 - 4*q^10 + 5*q^13 - 6*q^16 + 9*q^19 - ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] QPochhammer[ x^19, x^38])^2, {x, 0, n}]; (* Michael Somos, Oct 30 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^19 + A) / (eta(x^2 + A) * eta(x^38 + A)))^2, n))};

Crossrefs

Cf. A112199, A134004.

Sequence in context: A365005 A022597 A073252 * A132320 A353400 A076369

Adjacent sequences: A134002 A134003 A134004 * A134006 A134007 A134008

Keyword

sign

Author

Michael Somos, Oct 01 2007

Status

approved