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A202836 Expansion of e.g.f.: exp(9*x/(1-4*x)) / sqrt(1-16*x^2).

%I A202836 #16 Jun 23 2022 07:01:09
%S A202836 1,9,169,3969,119025,4173849,169754841,7764958161,395853630561,
%T A202836 22158814509225,1352182116776841,89167147951863969,
%U A202836 6319166996322943569,478498255838869322169,38549853656690487255225,3290600595687160597292529,296613603422471046790496961
%N A202836 Expansion of e.g.f.: exp(9*x/(1-4*x)) / sqrt(1-16*x^2).
%H A202836 G. C. Greubel, <a href="/A202836/b202836.txt">Table of n, a(n) for n = 0..350</a>
%F A202836 a(n) = A202837(n)^2, where the e.g.f. of A202837 is exp(3*x + 2*x^2).
%F A202836 a(n) ~ n^n*exp(3*sqrt(n)-9/8-n)*2^(2*n-1) * (1+33/(32*sqrt(n))). - _Vaclav Kotesovec_, May 23 2013
%F A202836 D-finite with recurrence: a(n) = (4*n+5)*a(n-1) + 4*(n-1)*(4*n+5)*a(n-2) - 64*(n-1)*(n-2)^2*a(n-3). - _Vaclav Kotesovec_, May 23 2013
%e A202836 E.g.f.: A(x) = 1 + 9*x + 169*x^2/2! + 3969*x^3/3! + 119025*x^4/4! + ...
%e A202836 where A(x) = 1 + 3^2*x + 13^2*x^2/2! + 63^2*x^3/3! + 345^2*x^4/4! + 2043^2*x^5/5! + ... + A202837(n)^2*x^n/n! + ...
%t A202836 CoefficientList[Series[Exp[9*x/(1-4*x)]/Sqrt[1-16*x^2], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, May 23 2013 *)
%o A202836 (PARI) {a(n)=n!*polcoeff(exp(9*x/(1-4*x)+x*O(x^n))/sqrt(1-16*x^2+x*O(x^n)),n)}
%o A202836 (PARI) {a(n)=n!^2*polcoeff(exp(3*x+2*x^2+x*O(x^n)),n)^2}
%o A202836 (PARI) {a(n)=sum(k=0,n\2,3^(n-2*k)*2^k*n!/((n-2*k)!*k!))^2}
%o A202836 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(9*x/(1-4*x))/Sqrt(1-16*x^2) ))); // _G. C. Greubel_, Jun 22 2022
%o A202836 (SageMath)
%o A202836 def A202836_list(prec):
%o A202836     P.<x> = PowerSeriesRing(QQ, prec)
%o A202836     return P( exp(9*x/(1-4*x))/sqrt(1-16*x^2) ).egf_to_ogf().list()
%o A202836 A202836_list(40) # _G. C. Greubel_, Jun 22 2022
%Y A202836 Cf. A202837, A202827, A202828, A202829, A202831, A202833, A202835.
%K A202836 nonn
%O A202836 0,2
%A A202836 _Paul D. Hanna_, Dec 25 2011