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

0,2

Comments

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

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

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

Example

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 + ...

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 *)

Prog

(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))};

Crossrefs

Cf. A138521, A210458, A223903.

Sequence in context: A031471 A045206 A138521 * A022600 A067237 A196157

Adjacent sequences: A225698 A225699 A225700 * A225702 A225703 A225704

Keyword

nonn

Author

Michael Somos, May 17 2013

Status

approved