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A202833 Expansion of e.g.f.: exp(9*x/(1-x)) / sqrt(1-x^2).

%I A202833 #22 Jun 23 2022 07:06:22
%S A202833 1,9,100,1296,19044,311364,5588496,108993600,2291345424,51585311376,
%T A202833 1236953249856,31447331115264,844332494760000,23859653712215616,
%U A202833 707522071322329344,21958125453144843264,711555574637600891136,24025060090437573945600
%N A202833 Expansion of e.g.f.: exp(9*x/(1-x)) / sqrt(1-x^2).
%H A202833 Vincenzo Librandi, <a href="/A202833/b202833.txt">Table of n, a(n) for n = 0..200</a>
%F A202833 a(n) = A202834(n)^2, where the e.g.f. of A202834 is exp(3*x + x^2/2).
%F A202833 a(n) = ( Sum_{k=0..[n/2]} 3^(n-2*k)/2^k * n!/((n-2*k)!*k!) )^2.
%F A202833 a(n) ~ n^n*exp(6*sqrt(n)-9/2-n)/2 * (1+15/(4*sqrt(n))). - _Vaclav Kotesovec_, May 23 2013
%F A202833 D-finite with recurrence: a(n) = (n+8)*a(n-1) + (n-1)*(n+8)*a(n-2) - (n-1)*(n-2)^2*a(n-3). - _Vaclav Kotesovec_, May 23 2013
%e A202833 E.g.f.: A(x) = 1 + 9*x + 100*x^2/2! + 1296*x^3/3! + 19044*x^4/4! + ...
%e A202833 where A(x) = 1 + 3^2*x + 10^2*x^2/2! + 36^2*x^3/3! + 138^2*x^4/4! + ... + A202834(n)^2*x^n/n! + ...
%t A202833 With[{nn=20},CoefficientList[Series[Exp[((9x)/(1-x))]/Sqrt[1-x^2], {x,0,nn}], x]Range[0,nn]!] (* _Harvey P. Dale_, Apr 23 2012 *)
%o A202833 (PARI) {a(n)=n!*polcoeff(exp(9*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}
%o A202833 (PARI) {a(n)=n!^2*polcoeff(exp(3*x+x^2/2+x*O(x^n)),n)^2}
%o A202833 (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(9*x/(1-x))/Sqrt(1-x^2) ))); // _G. C. Greubel_, Jun 22 2022
%o A202833 (SageMath)
%o A202833 def A202833_list(prec):
%o A202833     P.<x> = PowerSeriesRing(QQ, prec)
%o A202833     return P( exp(9*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
%o A202833 A202833_list(40) # _G. C. Greubel_, Jun 22 2022
%Y A202833 Cf. A202834, A202827, A202828, A202829, A202831, A202835, A202836.
%K A202833 nonn
%O A202833 0,2
%A A202833 _Paul D. Hanna_, Dec 25 2011