A058279 a(0)=a(1)=1, a(n)=a(n-2)+(n+1)*a(n-1).

1, 1, 4, 17, 89, 551, 3946, 32119, 293017, 2962289, 32878196, 397500641, 5200386529, 73202912047, 1103244067234, 17725107987791, 302430079859681, 5461466545462049, 104070294443638612, 2086867355418234289, 43928284758226558681, 968509132036402525271

Offset: 0

Author

N. J. A. Sloane, Dec 09 2000

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Chris Cannings, The Stationary Distributions of a Class of Markov Chains, Applied Mathematics, Vol. 4 No. 5, 2013, pp. 769-773.
• David Ji, Michael Li, and Daniel Wang, Parallel chip-firing games on directed graphs, arXiv:2407.15889 [math.CO], 2024.

Crossrefs

See A058307 for the same recurrence with 0,1 inputs. [Wolfdieter Lang, May 19 2010]

Programs

Magma

[n le 2 select 1 else Self(n-2)+Self(n-1)*(n): n in [1..30]]; // Vincenzo Librandi, May 06 2013

Maple

A058279 := proc(n) option remember; if n <= 1 then 1 else A058279(n-2)+(n+1)*A058279(n-1); fi; end;

Mathematica

RecurrenceTable[{a[0] == a[1] == 1, a[n] == a[n-2] + a[n-1] (n+1)}, a, {n, 30}] (* Vincenzo Librandi, May 06 2013 *)

Formula

a(n) is asymptotic to c*n! with c=0.9007... - Benoit Cloitre, Sep 03 2002

Right asymptotic (with offset=0) is a(n) ~ c * (n+1)!, where c = 2*BesselI(1,2)-BesselI(0,2) = 0.9016884069385908593273044... - Vaclav Kotesovec, Jan 05 2013

E.g.f.: 2*Pi*(I*BesselY(3, 2*I)*BesselI(2, 2*sqrt(1-x)) + BesselI(3, 2)*BesselY(2, 2*I*sqrt(1-x)))/(1-x). Such e.g.f. computations were inspired after e-mail exchange with Gary Detlefs. After differentiation and putting x=0 one has to use simplifications. See the Abramowitz-Stegun handbook, p.360, 9.1.16. [Wolfdieter Lang, May 19 2010]