A202835 Expansion of e.g.f.: exp(9*x/(1-2*x)) / sqrt(1-4*x^2).

1, 9, 121, 2025, 40401, 927369, 24000201, 689220009, 21710549025, 743187098889, 27441452694681, 1086166287819369, 45846179189949681, 2054407698719865225, 97357866191666622441, 4862830945258077841449, 255239441235423753980481, 14040944744510973314880009

Offset

0,2

Links

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

Formula

a(n) = A083886(n)^2, where the e.g.f. of A083886 is exp(3*x + x^2).

a(n) = ( Sum_{k=0..[n/2]} 3^(n-2*k) * n!/((n-2*k)!*k!) )^2.

a(n) ~ n^n*exp(3*sqrt(2*n)-9/4-n)*2^(n-1). - Vaclav Kotesovec, May 23 2013

D-finite with recurrence: a(n) = (2*n+7)*a(n-1) + 2*(n-1)*(2*n+7)*a(n-2) - 8*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

Example

E.g.f.: A(x) = 1 + 9*x + 121*x^2/2! + 2025*x^3/3! + 40401*x^4/4! +...
where A(x) = 1 + 3^2*x + 11^2*x^2/2! + 45^2*x^3/3! + 201^2*x^4/4! + 963^2*x^5/5! +...+ A083886(n)^2*x^n/n! +...

Mathematica

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

Prog

(PARI) {a(n)=n!*polcoeff(exp(9*x/(1-2*x)+x*O(x^n))/sqrt(1-4*x^2+x*O(x^n)), n)}
(PARI) {a(n)=n!^2*polcoeff(exp(3*x+x^2+x*O(x^n)), n)^2}
(PARI) {a(n)=sum(k=0, n\2, 3^(n-2*k)*n!/((n-2*k)!*k!))^2}
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(9*x/(1-2*x))/Sqrt(1-4*x^2) ))); // G. C. Greubel, Jun 21 2022
(SageMath)
def A202835_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( exp(9*x/(1-2*x))/sqrt(1-4*x^2) ).egf_to_ogf().list()
A202835_list(40) # G. C. Greubel, Jun 21 2022

Crossrefs

Cf. A083886, A202827, A202828, A202829, A202831, A202833, A202836.

Sequence in context: A046184 A084769 A246467 * A321847 A050353 A112941

Adjacent sequences: A202832 A202833 A202834 * A202836 A202837 A202838

Keyword

nonn

Author

Paul D. Hanna, Dec 25 2011

Status

approved