A202835 - OEIS

%I #20 Jun 22 2022 02:54:05

%S 1,9,121,2025,40401,927369,24000201,689220009,21710549025,

%T 743187098889,27441452694681,1086166287819369,45846179189949681,

%U 2054407698719865225,97357866191666622441,4862830945258077841449,255239441235423753980481,14040944744510973314880009

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

%H G. C. Greubel, <a href="/A202835/b202835.txt">Table of n, a(n) for n = 0..350</a>

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

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

%F a(n) ~ n^n*exp(3*sqrt(2*n)-9/4-n)*2^(n-1). - _Vaclav Kotesovec_, May 23 2013

%F 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

%e E.g.f.: A(x) = 1 + 9*x + 121*x^2/2! + 2025*x^3/3! + 40401*x^4/4! +...

%e 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! +...

%t 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 *)

%o (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)}

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

%o (PARI) {a(n)=sum(k=0,n\2,3^(n-2*k)*n!/((n-2*k)!*k!))^2}

%o (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

%o (SageMath)

%o def A202835_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( exp(9*x/(1-2*x))/sqrt(1-4*x^2) ).egf_to_ogf().list()

%o A202835_list(40) # _G. C. Greubel_, Jun 21 2022

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 25 2011