A134004 Expansion of (chi(-x) * chi(-x^19))^(-2) in powers of x where chi() is a Ramanujan theta function.

1, 2, 3, 6, 9, 14, 22, 32, 46, 66, 93, 128, 176, 238, 319, 426, 562, 736, 960, 1244, 1602, 2054, 2620, 3324, 4203, 5292, 6634, 8290, 10322, 12808, 15845, 19542, 24028, 29468, 36042, 43966, 53506, 64960, 78685, 95106, 114709, 138066, 165855, 198856, 237979

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..44.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-5/3) * (eta(q^2) * eta(q^38) / (eta(q) * eta(q^19)))^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)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134005.

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

Convolution inverse of A134005.

a(n) ~ exp(2*Pi*sqrt(10*n/57)) * 5^(1/4) / (2^(11/4) * 57^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

Example

G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 9*x^4 + 14*x^5 + 22*x^6 + 32*x^7 + 46*x^8 + ...
G.f. = q^5 + 2*q^8 + 3*q^11 + 6*q^14 + 9*q^17 + 14*q^20 + 22*q^23 + 32*q^26 + ...

Mathematica

nmax = 40; CoefficientList[Series[Product[((1 + x^k) * (1 + x^(19*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] QPochhammer[ -x^19, x^19])^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^2 + A) * eta(x^38 + A) / (eta(x + A) * eta(x^19 + A)))^2, n))};

Crossrefs

Cf. A134005.

Sequence in context: A333697 A128518 A022567 * A123631 A228364 A018060

Adjacent sequences: A134001 A134002 A134003 * A134005 A134006 A134007

Keyword

nonn

Author

Michael Somos, Oct 01 2007

Status

approved