A053487 E.g.f.: exp(4x)/(1-x).

1, 5, 26, 142, 824, 5144, 34960, 261104, 2154368, 19651456, 197563136, 2177388800, 26145442816, 339957865472, 4759678552064, 71396252022784, 1142344327331840, 19419870744510464, 349557742120665088, 6641597375170543616, 132831948602922500096

Offset: 0

Author

N. J. A. Sloane, Jan 15 2000

Keywords

• nonn

Comments

a(n) is the binomial transform of A053486. More generally, for every integer k, the sequence whose e.g.f is exp((k+1)*x)/(1-x) is the binomial transform of the sequence whose e.g.f is exp(k*x)/(1-x). - Groux Roland, Mar 23 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Programs

Maple

F(x) := exp(4*x)/(1-x): f[0]:=F(x): for n from 1 to 20 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20); # Zerinvary Lajos, Apr 03 2009
seq(simplify(KummerU(-n, -n, 4)), n = 0..20); # Peter Luschny, May 10 2022

Mathematica

With[{nn=30},CoefficientList[Series[Exp[4x]/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 09 2013 *)

Formula

a(n) is the permanent of the n X n matrix with 5's on the diagonal and 1's elsewhere. a(n) = Sum_{k=0..n} A008290(n, k)*5^k. - Philippe Deléham, Dec 12 2003

E.g.f.: exp(4x)/(1-x)=1/E(0); E(k)=1-x/(1-4/(4+(k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011

G.f.: 1/Q(0), where Q(k)= 1 - 4*x - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013

a(n) ~ n! * exp(4). - Vaclav Kotesovec, Jun 21 2013

a(n) = exp(4)*Gamma(n+1,4). - Gerry Martens, Jul 24 2015

a(n) = KummerU(-n, -n, 4). - Peter Luschny, May 10 2022