%I #24 Jun 22 2022 02:54:09
%S 1,4,25,196,1849,20164,249001,3422500,51739249,851822596,15155825881,
%T 289527934084,5906625426025,128089110981316,2940882813228649,
%U 71239270847432164,1815115761586307041,48511703775281296900,1356708799439194070809,39615996090901693902916
%N Expansion of e.g.f.: exp(4*x/(1-x)) / sqrt(1-x^2).
%H Vincenzo Librandi, <a href="/A202827/b202827.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = A005425(n)^2, where the e.g.f. of A005425 is exp(2*x + x^2/2).
%F a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*n!/((n-2*k)!*k!) )^2. [From formula by Huajun Huang in A005425]
%F a(n) ~ n^n*exp(4*sqrt(n)-2-n)/2 * (1+5/(3*sqrt(n))). - _Vaclav Kotesovec_, May 23 2013
%F D-finite with recurrence: a(n) = (n+3)*a(n-1) +(n-1)*(n+3)*a(n-2) - (n-1)*(n-2)^2*a(n-3). - _Vaclav Kotesovec_, May 23 2013
%e E.g.f.: A(x) = 1 + 4*x + 25*x^2/2! + 196*x^3/3! + 1849*x^4/4! +...
%e where A(x) = 1 + 2^2*x + 5^2*x^2/2! + 14^2*x^3/3! + 43^2*x^4/4! +...+ A005425(n)^2*x^n/n! +...
%t With[{nn=20},CoefficientList[Series[Exp[(4x)/(1-x)]/Sqrt[1-x^2], {x,0,nn}], x]Range[0,nn]!] (* _Harvey P. Dale_, Dec 31 2011 *)
%o (PARI) {a(n)=n!*polcoeff(exp(4*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}
%o (PARI) {a(n)=sum(k=0,n\2,2^(n-3*k)*n!/((n-2*k)!*k!))^2}
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(4*x/(1-x))/Sqrt(1-x^2) ))); // _G. C. Greubel_, Jun 21 2022
%o (SageMath)
%o def A202827_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( exp(4*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
%o A202827_list(40) # _G. C. Greubel_, Jun 21 2022
%Y Cf. A005425, A202828, A202829, A202831, A202833, A202835, A202836.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 25 2011
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