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%I #49 May 10 2022 15:19:10
%S 1,5,27,159,1029,7353,58095,506691,4860297,51023277,583097859,
%T 7215769575,96210083853,1375803720801,21012273704151,341449444105227,
%U 5883436565417745,107162594556721749,2057521815411573483
%N E.g.f.: exp(3*x)/(1-x)^2.
%C Binomial transform of A081923
%C Polynomials in A010027 evaluated at 4. - _Ralf Stephan_, Dec 15 2004
%H Iain Fox, <a href="/A081924/b081924.txt">Table of n, a(n) for n = 0..450</a> (first 200 terms from Vincenzo Librandi)
%F E.g.f.: exp(3*x)/(1-x)^2.
%F Define f_1(x), f_2(x), ... such that f_1(x) = x*e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n-1) = e^{-1/2}*2^n*f_n(1/2). - _Milan Janjic_, May 30 2008
%F G.f.: hypergeom([1,2],[],x/(1-3*x))/(1-3*x). - _Mark van Hoeij_, Nov 08 2011
%F a(n) + (-n-4)*a(n-1) + 3*(n-1)*a(n-2) = 0. - _R. J. Mathar_, Nov 24 2012
%F G.f.: 2/x/G(0) - 1/x, where G(k) = 1 + 1/(1 - x*(2*k+2)/(x*(2*k+5) - 1 + x*(2*k+2)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 31 2013
%F G.f.: G(0)/x - 1/x, where G(k) = 1 + (2*k + 1)*x/(1-3*x - 2*x*(1-3*x)*(k+1)/(2*x*(k+1) + (1-3*x)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 26 2013
%F E.g.f.: 1/E(0), where E(k) = 1 - 2*x/(1 - x/(x - 2 + 6/(3 - x*(k+1)/E(k+1)))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 26 2013
%F G.f.: (Sum_{k>=0} (k!*(x/(1-3*x))^k) - 1)/x = Q(0)/(2*x) - 1/x, where Q(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + (1-3*x)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Aug 09 2013
%F a(n) ~ exp(3) * n! * n. - _Vaclav Kotesovec_, Oct 05 2013
%F From _Peter Bala_, Jul 22 2021: (Start)
%F a(n) = n!*Sum_{k = 0..n} (n - k + 1)*3^k/k!.
%F a(n) = (n+1)!*hypergeom([-n],[-n-1],3).
%F (n-2)*a(n+1) = (n^2-1)*a(n) - 3^(n+2). (End)
%F a(n) = KummerU(-n, -n - 1, 3). - _Peter Luschny_, May 10 2022
%p seq(simplify(KummerU(-n, -n - 1, 3)), n = 0..20); # _Peter Luschny_, May 10 2022
%t With[{nn=20},CoefficientList[Series[Exp[3x]/(1-x)^2,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 14 2013 *)
%o (PARI) my(x='x+O('x^66)); Vec( serlaplace(exp(3*x)/(1-x)^2) ) \\ _Joerg Arndt_, Aug 15 2013
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 01 2003
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