A052898 2*n! + 1.

3, 3, 5, 13, 49, 241, 1441, 10081, 80641, 725761, 7257601, 79833601, 958003201, 12454041601, 174356582401, 2615348736001, 41845579776001, 711374856192001, 12804747411456001, 243290200817664001

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]]; /* or */ [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

Cf. sequences of the type k*n!+1: A038507 (k=1), this sequence, A173324 (k=3), A173322 (k=4), A173319 (k=5), A173314 (k=6).

Sequence in context: A367005 A226921 A133190 * A183483 A218663 A095355

Adjacent sequences: A052895 A052896 A052897 * A052899 A052900 A052901

Keyword

nonn,easy

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Extensions

Definition replaced with the closed formula by Bruno Berselli, Sep 28 2013

Status

approved