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%I #19 Jun 23 2022 07:01:25
%S 1,16,289,5776,126025,2972176,75186241,2027520784,57988974481,
%T 1751546371600,55668326576641,1855807478279056,64713593898036889,
%U 2354701531657512976,89209297718289390625,3512141211682081889296,143435878498076017059361
%N Expansion of e.g.f.: exp(16*x/(1-x)) / sqrt(1-x^2).
%H G. C. Greubel, <a href="/A202878/b202878.txt">Table of n, a(n) for n = 0..350</a>
%F a(n) = A202879(n)^2, where the e.g.f. of A202879 is exp(4*x + x^2/2).
%F a(n) = ( Sum_{k=0..floor(n/2)} 4^(n-2*k)/2^k * n!/((n-2*k)!*k!) )^2.
%F a(n) ~ n^n*exp(8*sqrt(n)-8-n)/2 * (1+22/(3*sqrt(n))). - _Vaclav Kotesovec_, May 23 2013
%F D-finite with recurrence: a(n) = (n+15)*a(n-1) + (n-1)*(n+15)*a(n-2) - (n-1)*(n-2)^2*a(n-3). - _Vaclav Kotesovec_, May 23 2013
%e E.g.f.: A(x) = 1 + 16*x + 289*x^2/2! + 5776*x^3/3! + 126025*x^4/4! + ...
%e where A(x) = 1 + 4^2*x + 17^2*x^2/2! + 76^2*x^3/3! + 355^2*x^4/4! + 1724^2*x^5/5! + ... + A202879(n)^2*x^n/n! + ...
%t CoefficientList[Series[Exp[16*x/(1-x)]/Sqrt[1-x^2], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, May 23 2013 *)
%o (PARI) {a(n)=n!*polcoeff(exp(16*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}
%o (PARI) {a(n)=n!^2*polcoeff(exp(4*x+x^2/2+x*O(x^n)),n)^2}
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(16*x/(1-x))/Sqrt(1-x^2) ))); // _G. C. Greubel_, Jun 22 2022
%o (SageMath)
%o def A202878_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( exp(16*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
%o A202878_list(40) # _G. C. Greubel_, Jun 22 2022
%Y Cf. A202879, A202827, A202828, A202829, A202831, A202835, A202836, A202837.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 25 2011
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