A139631 Expansion of chi(x^5) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function.

1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 3, 2, 4, 2, 5, 4, 6, 5, 8, 6, 11, 8, 13, 10, 16, 14, 20, 17, 24, 21, 31, 26, 37, 32, 44, 41, 54, 49, 64, 59, 79, 72, 94, 86, 111, 106, 132, 126, 156, 149, 187, 178, 219, 210, 257, 251, 302, 295, 352, 346, 416, 406, 483, 474, 560

Offset: 0

Author

Michael Somos, Apr 27 2008

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + x^2 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + 3*x^10 + 2*x^11 + ...
G.f. = 1/q + q^15 + q^31 + q^39 + 2*q^47 + q^55 + 2*q^63 + q^71 + 3*q^79 + ...
		

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. A139632.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^2] QPochhammer[ -x^5, x^10], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k)) * (1 + x^(5*k)) / (1 + x^(10*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^10 + A)^2 / (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};

Formula

Expansion of q^(1/8) * eta(q^4) * eta(q^10)^2 / (eta(q^2) * eta(q^5) * eta(q^20)) in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139632.

G.f.: Product_{k>0} (1 + x^(2*k)) * (1 + x^(5*k)) / (1 + x^(10*k)).

a(n) = A139632(2*n).

a(n) ~ exp(Pi*sqrt(n/5)) / (4 * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015