A202829 - OEIS

%I #19 Jun 22 2022 02:54:18

%S 1,4,49,676,13225,293764,7890481,236359876,8052729169,300797402500,

%T 12388985000401,551925653637604,26614517015830969,1373655853915667716,

%U 75803216516463190225,4440662493517062816004,275697752917311709134241,18052104090118575573856516

%N Expansion of e.g.f.: exp(4*x/(1-3*x)) / sqrt(1-9*x^2).

%H G. C. Greubel, <a href="/A202829/b202829.txt">Table of n, a(n) for n = 0..350</a>

%F a(n) = A202830(n)^2, where the e.g.f. of A202830 is exp(2*x + 3*x^2/2).

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

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

%F D-finite with recurrence: a(n) = (3*n+1)*a(n-1) + 3*(n-1)*(3*n+1)*a(n-2) - 27*(n-1)*(n-2)^2*a(n-3). - _Vaclav Kotesovec_, May 23 2013

%e E.g.f.: A(x) = 1 + 4*x + 49*x^2/2! + 676*x^3/3! + 13225*x^4/4! + 293764*x^5/5! + ...

%e were A(x) = 1 + 2^2*x + 7^2*x^2/2! + 26^2*x^3/3! + 115^2*x^4/4! + 542^2*x^5/5! + ... + A202830(n)^2*x^n/n! + ...

%t With[{nn=20},CoefficientList[Series[Exp[(4x)/(1-3x)]/Sqrt[1-9x^2],{x,0,nn}], x]Range[0,nn]!] (* _Harvey P. Dale_, Mar 09 2012 *)

%o (PARI) {a(n)=n!*polcoeff(exp(4*x/(1-3*x)+x*O(x^n))/sqrt(1-9*x^2+x*O(x^n)),n)}

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

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(4*x/(1-3*x))/Sqrt(1-9*x^2) ))); // _G. C. Greubel_, Jun 21 2022

%o (SageMath)

%o def A202829_list(prec):

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

%o     return P( exp(4*x/(1-3*x))/sqrt(1-9*x^2) ).egf_to_ogf().list()

%o A202829_list(40) # _G. C. Greubel_, Jun 21 2022

%Y Cf. A202830, A202827, A202828, A202831, A202833, A202835, A202836.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 25 2011