OFFSET
0,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 874
FORMULA
E.g.f.: (-2-exp(x)+x*exp(x))/(-1+x).
Recurrence: {a(2)=5, a(1)=3, (n^2+2*n+1)*a(n)+(-n^2-3*n-1)*a(n+1)+a(n+2)*n}
From Sergei N. Gladkovskii, Jul 04 2012: (Start)
a(0)=3; for n>0, a(n) = n*a(n-1)-n+1.
Let E(x) be the e.g.f., then
E(x)=(x*G(0)-2)/(x-1), where G(k)= 1 - 1/(x - x^3/(x^2 - (k+1)/G(k+1)));(continued fraction, 3rd kind, 3-step).
E(x)=x*G(0)/(x-1), where G(k)= 1 - 1/(x + 2*x*(x-1)*k!/(1 - 2*(x-1)*k! - x^2/(x^2 + 2*(x-1)*(k+1)!/G(k+1)))); (continued fraction, 3rd kind, 4-step).
(End).
MAPLE
spec := [S, {S=Union(Sequence(Z), Sequence(Z), Set(Z))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
a[0]:=3: for n from 1 to 21 do a[n]:=n*a[n-1]-n+1; od:
seq(a[n], n=0..20). # Sergei N. Gladkovskii, Jul 04 2012
MATHEMATICA
lst={}; s=-3; Do[s+=(n+=s*n); AppendTo[lst, s], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 10 2008 *)
FoldList[#1*#2 - #2 + 1 &, 3, Range[19]] (* Robert G. Wilson v, Jul 07 2012 *)
Table[2 n! + 1, {n, 0, 20}] (* Vincenzo Librandi, Sep 29 2013 *)
PROG
(Magma) [2*Factorial(n) + 1: n in [0..20]]; // Vincenzo Librandi, Sep 29 2013
(Magma) [3] cat [n eq 1 select n+2 else n*Self(n-1)-n+1: n in [1..25] ]; // Vincenzo Librandi, Sep 29 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000
EXTENSIONS
Definition replaced with the closed formula by Bruno Berselli, Sep 28 2013
STATUS
approved