A225701 Expansion of chi(q)^5 / chi(q^5) in powers of q where chi() is a Ramanujan theta function.

1, 5, 10, 15, 30, 55, 80, 120, 190, 285, 410, 585, 840, 1190, 1640, 2240, 3070, 4170, 5570, 7400, 9830, 12960, 16920, 21990, 28520, 36805, 47180, 60225, 76720, 97350, 122880, 154610, 194110, 242880, 302740, 376295, 466710, 577270, 711800, 875520, 1074790

Offset: 0

Author

Michael Somos, May 17 2013

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 5*q + 10*q^2 + 15*q^3 + 30*q^4 + 55*q^5 + 80*q^6 + 120*q^7 + 190*q^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A138521, A210458, A223903.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^5 / QPochhammer[ -q^5, q^10], {q, 0, n}];
nmax=60; CoefficientList[Series[Product[(1-x^k)^5 * (1+x^k)^10 * (1+x^(10*k)) / ((1-x^(4*k))^5 * (1+x^(5*k))),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)

PARI

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

Formula

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

Euler transform of period 20 sequence [ 5, -5, 5, 0, 4, -5, 5, 0, 5, -4, 5, 0, 5, -5, 4, 0, 5, -5, 5, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. of A223903.

a(n) = (-1)^n * A138521(n). a(n) = 5 * A210458(n) unless n=0.

a(n) ~ exp(2*Pi*sqrt(n/5)) / (2 * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

Empirical: Sum_{n>=0} a(n)/exp(Pi*n) = sqrt(5) - 1. - Simon Plouffe, Mar 02 2021