A080954 E.g.f. exp(5x)/(1-x).

1, 6, 37, 236, 1569, 10970, 81445, 648240, 5576545, 52142030, 531185925, 5891873300, 70946620225, 923526766050, 12935478240325, 194062691183000, 3105155646818625, 52788408935369750, 950195175533921125

Offset: 0

Author

Paul Barry, Feb 26 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A053487. 4th Binomial transform of A000522. Fifth binomial transform of n! = A000142.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A008290, A010842, A053486.

Programs

Maple

F(x):= exp(5*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..18); # Zerinvary Lajos, Apr 03 2009

Mathematica

With[{nn=20},CoefficientList[Series[Exp[5x]/(1-x),{x,0,nn}],x] Range[0, nn]!] (* Harvey P. Dale, Sep 19 2011 *)
Table[n!*Sum[5^k/k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 28 2012 *)
RecurrenceTable[{a[0]==1, a[n]==n*a[n-1] + 5^n}, a, {n, 20}] (* Vincenzo Librandi, Nov 15 2012 *)

Formula

a(n) = n! * Sum_{k=0..n} 5^k/k!.

a(n) is the permanent of the n X n matrix with 6's on the diagonal and 1's elsewhere. a(n) = Sum_{k=0..n} A008290(n, k)*6^k. - Philippe Deléham, Dec 12 2003

Conjecture: -a(n) + (n+5)*a(n-1) + 5*(1-n)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011

a(n) ~ n!*exp(5). - Vaclav Kotesovec, Oct 13 2012

a(0)=0, a(n) = n * a(n-1) + 5^n. - Vincenzo Librandi, Nov 15 2012