A202836 Expansion of e.g.f.: exp(9*x/(1-4*x)) / sqrt(1-16*x^2).

1, 9, 169, 3969, 119025, 4173849, 169754841, 7764958161, 395853630561, 22158814509225, 1352182116776841, 89167147951863969, 6319166996322943569, 478498255838869322169, 38549853656690487255225, 3290600595687160597292529, 296613603422471046790496961

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..350

Formula

a(n) = A202837(n)^2, where the e.g.f. of A202837 is exp(3*x + 2*x^2).

a(n) ~ n^n*exp(3*sqrt(n)-9/8-n)*2^(2*n-1) * (1+33/(32*sqrt(n))). - Vaclav Kotesovec, May 23 2013

D-finite with recurrence: a(n) = (4*n+5)*a(n-1) + 4*(n-1)*(4*n+5)*a(n-2) - 64*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

Example

E.g.f.: A(x) = 1 + 9*x + 169*x^2/2! + 3969*x^3/3! + 119025*x^4/4! + ...
where A(x) = 1 + 3^2*x + 13^2*x^2/2! + 63^2*x^3/3! + 345^2*x^4/4! + 2043^2*x^5/5! + ... + A202837(n)^2*x^n/n! + ...

Mathematica

CoefficientList[Series[Exp[9*x/(1-4*x)]/Sqrt[1-16*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

Prog

(PARI) {a(n)=n!*polcoeff(exp(9*x/(1-4*x)+x*O(x^n))/sqrt(1-16*x^2+x*O(x^n)), n)}
(PARI) {a(n)=n!^2*polcoeff(exp(3*x+2*x^2+x*O(x^n)), n)^2}
(PARI) {a(n)=sum(k=0, n\2, 3^(n-2*k)*2^k*n!/((n-2*k)!*k!))^2}
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(9*x/(1-4*x))/Sqrt(1-16*x^2) ))); // G. C. Greubel, Jun 22 2022
(SageMath)
def A202836_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( exp(9*x/(1-4*x))/sqrt(1-16*x^2) ).egf_to_ogf().list()
A202836_list(40) # G. C. Greubel, Jun 22 2022

Crossrefs

Cf. A202837, A202827, A202828, A202829, A202831, A202833, A202835.

Sequence in context: A017306 A210089 A243949 * A052774 A276960 A122725

Adjacent sequences: A202833 A202834 A202835 * A202837 A202838 A202839

Keyword

nonn

Author

Paul D. Hanna, Dec 25 2011

Status

approved