This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A080954 #37 Aug 23 2024 08:40:17
%S A080954 1,6,37,236,1569,10970,81445,648240,5576545,52142030,531185925,
%T A080954 5891873300,70946620225,923526766050,12935478240325,194062691183000,
%U A080954 3105155646818625,52788408935369750,950195175533921125
%N A080954 E.g.f. exp(5x)/(1-x).
%C A080954 Binomial transform of A053487. 4th Binomial transform of A000522. Fifth binomial transform of n! = A000142.
%H A080954 Vincenzo Librandi, <a href="/A080954/b080954.txt">Table of n, a(n) for n = 0..200</a>
%F A080954 a(n) = n! * Sum_{k=0..n} 5^k/k!.
%F A080954 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
%F A080954 Conjecture: -a(n) + (n+5)*a(n-1) + 5*(1-n)*a(n-2) = 0. - _R. J. Mathar_, Nov 14 2011
%F A080954 a(n) ~ n!*exp(5). - _Vaclav Kotesovec_, Oct 13 2012
%F A080954 a(0)=0, a(n) = n * a(n-1) + 5^n. - _Vincenzo Librandi_, Nov 15 2012
%p A080954 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
%t A080954 With[{nn=20},CoefficientList[Series[Exp[5x]/(1-x),{x,0,nn}],x] Range[0, nn]!] (* _Harvey P. Dale_, Sep 19 2011 *)
%t A080954 Table[n!*Sum[5^k/k!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 28 2012 *)
%t A080954 RecurrenceTable[{a[0]==1, a[n]==n*a[n-1] + 5^n}, a, {n, 20}] (* _Vincenzo Librandi_, Nov 15 2012 *)
%Y A080954 Cf. A008290, A010842, A053486.
%K A080954 easy,nonn
%O A080954 0,2
%A A080954 _Paul Barry_, Feb 26 2003