A081924 E.g.f.: exp(3*x)/(1-x)^2.
1, 5, 27, 159, 1029, 7353, 58095, 506691, 4860297, 51023277, 583097859, 7215769575, 96210083853, 1375803720801, 21012273704151, 341449444105227, 5883436565417745, 107162594556721749, 2057521815411573483
Offset: 0
Links
- Iain Fox, Table of n, a(n) for n = 0..450 (first 200 terms from Vincenzo Librandi)
Programs
-
Maple
seq(simplify(KummerU(-n, -n - 1, 3)), n = 0..20); # Peter Luschny, May 10 2022
-
Mathematica
With[{nn=20},CoefficientList[Series[Exp[3x]/(1-x)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 14 2013 *) -
PARI
my(x='x+O('x^66)); Vec( serlaplace(exp(3*x)/(1-x)^2) ) \\ Joerg Arndt, Aug 15 2013
Formula
E.g.f.: exp(3*x)/(1-x)^2.
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
G.f.: hypergeom([1,2],[],x/(1-3*x))/(1-3*x). - Mark van Hoeij, Nov 08 2011
a(n) + (-n-4)*a(n-1) + 3*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
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
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
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
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
a(n) ~ exp(3) * n! * n. - Vaclav Kotesovec, Oct 05 2013
From Peter Bala, Jul 22 2021: (Start)
a(n) = n!*Sum_{k = 0..n} (n - k + 1)*3^k/k!.
a(n) = (n+1)!*hypergeom([-n],[-n-1],3).
(n-2)*a(n+1) = (n^2-1)*a(n) - 3^(n+2). (End)
a(n) = KummerU(-n, -n - 1, 3). - Peter Luschny, May 10 2022
Comments