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

1,2

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 = 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

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

Example

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 + ...

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 *)

Prog

(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))}

Crossrefs

Cf. A138519, A145740.

Sequence in context: A351203 A138519 A138520 * A228864 A289434 A049794

Adjacent sequences: A210455 A210456 A210457 * A210459 A210460 A210461

Keyword

nonn

Author

Michael Somos, Jan 21 2013

Status

approved