A202833 Expansion of e.g.f.: exp(9*x/(1-x)) / sqrt(1-x^2).

1, 9, 100, 1296, 19044, 311364, 5588496, 108993600, 2291345424, 51585311376, 1236953249856, 31447331115264, 844332494760000, 23859653712215616, 707522071322329344, 21958125453144843264, 711555574637600891136, 24025060090437573945600

Offset: 0

Author

Paul D. Hanna, Dec 25 2011

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + 9*x + 100*x^2/2! + 1296*x^3/3! + 19044*x^4/4! + ...
where A(x) = 1 + 3^2*x + 10^2*x^2/2! + 36^2*x^3/3! + 138^2*x^4/4! + ... + A202834(n)^2*x^n/n! + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A202834, A202827, A202828, A202829, A202831, A202835, A202836.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(9*x/(1-x))/Sqrt(1-x^2) ))); // G. C. Greubel, Jun 22 2022

Mathematica

With[{nn=20},CoefficientList[Series[Exp[((9x)/(1-x))]/Sqrt[1-x^2], {x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Apr 23 2012 *)

PARI

{a(n)=n!*polcoeff(exp(9*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}

PARI

{a(n)=n!^2*polcoeff(exp(3*x+x^2/2+x*O(x^n)),n)^2}

SageMath

def A202833_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(9*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
A202833_list(40) # G. C. Greubel, Jun 22 2022

Formula

a(n) = A202834(n)^2, where the e.g.f. of A202834 is exp(3*x + x^2/2).

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

a(n) ~ n^n*exp(6*sqrt(n)-9/2-n)/2 * (1+15/(4*sqrt(n))). - Vaclav Kotesovec, May 23 2013

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