A210458 Expansion of q * (psi(-q^5) / psi(-q))^2 in powers of q where psi() is a Ramanujan theta function.

1, 2, 3, 6, 11, 16, 24, 38, 57, 82, 117, 168, 238, 328, 448, 614, 834, 1114, 1480, 1966, 2592, 3384, 4398, 5704, 7361, 9436, 12045, 15344, 19470, 24576, 30922, 38822, 48576, 60548, 75259, 93342, 115454, 142360, 175104, 214958, 263262, 321584, 391993, 476952

Offset: 1

Author

Michael Somos, Jan 21 2013

Keywords

• nonn

Comments

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

Examples

			q + 2*q^2 + 3*q^3 + 6*q^4 + 11*q^5 + 16*q^6 + 24*q^7 + 38*q^8 + 57*q^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..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. A138519, A145740.

Programs

Mathematica

nmax=60; CoefficientList[Series[Product[((1+x^k) * (1-x^(5*k)) * (1+x^(10*k)) / (1-x^(4*k)))^2,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)
a[n_]:= SeriesCoefficient[(EllipticTheta[2, 0, I*q^(5/2)]/EllipticTheta[ 2, 0, I*Sqrt[q]])^2, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Dec 07 2017 *)

PARI

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)))^2, n))}

Formula

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

Euler transform of period 20 sequence [ 2, 0, 2, 2, 0, 0, 2, 2, 2, 0, 2, 2, 2, 0, 0, 2, 2, 0, 2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (1 + u) * (1 + 5*u) * v * (1 + v) * (1 + 5*v).

G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is g.f. of A145740.

G.f.: x * (Product_{k>0} P(5, x^k) * P(20, x^k))^2 where P(n, x) is the n-th cyclotomic polynomial.

A138519(n) = -(-1)^n * a(n). Convolution inverse of A145740.

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

Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = -2/5 + (1/5)*sqrt(5). - Simon Plouffe, Mar 02 2021