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%I #19 Jun 22 2022 07:57:38
%S 1,4,81,1444,44521,1397124,58354321,2574344644,136043683281,
%T 7657406908804,489836445798001,33351743794661604,2504378700538997881,
%U 199445618093659242244,17189578072429077875121,1564487078400498014277124,152146464623361858013314721
%N Expansion of e.g.f.: exp(4*x/(1-5*x)) / sqrt(1-25*x^2).
%H Vincenzo Librandi, <a href="/A202831/b202831.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = A202832(n)^2, where the e.g.f. of A202832 is exp(2*x + 5*x^2/2).
%F a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*5^k * n!/((n-2*k)!*k!) )^2.
%F a(n) ~ n^n*exp(4*sqrt(n/5)-2/5-n)*5^n/2. - _Vaclav Kotesovec_, May 23 2013
%F D-finite with recurrence: a(n) = (5*n-1)*a(n-1) + 5*(n-1)*(5*n-1)*a(n-2) - 125*(n-1)*(n-2)^2*a(n-3). - _Vaclav Kotesovec_, May 23 2013
%e E.g.f.: 1 + 4*x + 81*x^2/2! + 1444*x^3/3! + 44521*x^4/4! + 1397124*x^5/5! + ...
%e where A(x) = 1 + 2^2*x + 9^2*x^2/2! + 38^2*x^3/3! + 211^2*x^4/4! + 1182^2*x^5/5! + ... + A202832(n)^2*x^n/n! + ...
%t CoefficientList[Series[Exp[4*x/(1-5*x)]/Sqrt[1-25*x^2], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, May 23 2013 *)
%o (PARI) {a(n)=n!*polcoeff(exp(4*x/(1-5*x)+x*O(x^n))/sqrt(1-25*x^2+x*O(x^n)),n)}
%o (PARI) {a(n)=n!^2*polcoeff(exp(2*x+5*x^2/2+x*O(x^n)),n)^2}
%o (PARI) {a(n)=sum(k=0,n\2,2^(n-3*k)*5^k*n!/((n-2*k)!*k!))^2}
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(4*x/(1-5*x))/Sqrt(1-25*x^2) ))); // _G. C. Greubel_, Jun 21 2022
%o (SageMath)
%o def A202831_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( exp(4*x/(1-5*x))/sqrt(1-25*x^2) ).egf_to_ogf().list()
%o A202831_list(40) # _G. C. Greubel_, Jun 21 2022
%Y Cf. A202832, A202827, A202828, A202829, A202833, A202835, A202836.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 25 2011
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