A058307 a(n) = (n+1)*a(n-1) + a(n-2), with a(0)=0, a(1)=1.

0, 1, 3, 13, 68, 421, 3015, 24541, 223884, 2263381, 25121075, 303716281, 3973432728, 55931774473, 842950049823, 13543132571641, 231076203767720, 4172914800390601, 79516457411189139, 1594502063024173381, 33564059780918830140, 740003817243238436461, 17053651856375402868743

Offset: 0

Author

N. J. A. Sloane, Dec 09 2000

Keywords

• nonn

Comments

Numerator of convergent to BesselI(0,2)/BesselI(1,2) for which the continued fraction expansion is [1,2,3....,n]. - Benoit Cloitre, Mar 27 2003

Numerator of continued fraction C(n) minus denominator of continued fraction C(n), where C(n) = [ 1; 2,3,4,...n ]. - Melvin Peralta, Jan 17 2017

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445 (terms n=0..100 from T. D. Noe)
• Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
• C. Cannings, The Stationary Distributions of a Class of Markov Chains, Applied Mathematics, Vol. 4 No. 5, 2013, pp. 769-773.
• S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.
• Russell Walsmith, Cl-Chemy II

Crossrefs

A column of A058294. Except for first term, -1 times row sums of A053495.

Cf. A001053, A060997, A058798.

Programs

GAP

a:= function(n)
    if n<2 then return n;
    else return (n+1)*a(n-1) + a(n-2);
    fi;
  end;
List([0..30], n-> a(n) ); # G. C. Greubel, Oct 07 2019

Magma

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

Maple

A058307 := proc(n) option remember; if n <= 1 then n else A058307(n-2)+(n+1)*A058307(n-1); fi; end;
a:= proc(n) option remember;
      if n<2 then n
    else (n+1)*a(n-1) + a(n-2)
      fi;
    end:
seq(a(n), n=0..30); # G. C. Greubel, Oct 07 2019

Mathematica

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(n+1)*a[n-1]+a[n-2]}, a, {n, 0, 30}] (* Vincenzo Librandi, May 06 2013 *)
Table[FullSimplify[(-BesselI[2+n,-2] * BesselK[2,2] + BesselI[2,2] * BesselK[2+n,2]) / (BesselI[3,2] * BesselK[2,2] + BesselI[2,2] * BesselK[3,2])],{n,0,20}] (* Vaclav Kotesovec, Feb 14 2014 *)
a[n_]:= a[n]= If[n<2, n, (n+1)*a[n-1] +a[n-2]]; Table[a[n], {n,0,30}] (* G. C. Greubel, Oct 07 2019 *)

PARI

my(m=30, v=concat([0,1], vector(m-2))); for(n=3, m, v[n]=n*v[n-1] +v[n-2]); v \\ G. C. Greubel, Nov 24 2018

Sage

def A058307(n):
    if n < 2: return n
    return factorial(n+1)*hypergeometric([1/2-n/2,1-n/2], [3,-1-n,1-n], 4)/2
[round(A058307(n).n(100)) for n in (0..21)] # Peter Luschny, Sep 10 2014

Sage

@CachedFunction
def a(n):
    if (n<2): return n
    else: return (n+1)*a(n-1) + a(n-2)
[a(n) for n in (0..30)]  # G. C. Greubel, Nov 24 2018

Formula

From Wouter Meeussen, Feb 02 2001: (Start)

a(2*r+1) = Sum_{j=0..r} (binomial(r+j, r-j)*(r+j)!/(r-j)! - binomial(r + j, r-j-1)*(r+j+1)!/(r-j)!) and

a(2*r) = Sum_{j=0..r} (binomial(r+j+1, r-j)*(r+j+1)!/(r-j)! - binomial(r +j, r-j)*(r+j+1)!/(r-j+1)! + binomial(r+j+1, r-j)*(r+j+1)!/(r-j)!). (End)

E.g.f.: Pi*(BesselI(2, 2)*BesselY(2, 2*I*sqrt(1-x)) - BesselY(2,2*I)*BesselI(2, 2*sqrt(1-x)))/(1-x). Motivated to look into e.g.f.'s for such recurrences by email exchange with Gary Detlefs. One has to use simplifications after differentiation and putting x=0. See Abramowitz-Stegun handbook p. 360, 9.1.16. - Wolfdieter Lang, May 18 2010

Limit n->infinity a(n)/(n+1)! = BesselI(0,2)-BesselI(1,2) = 0.688948447698738204... - Vaclav Kotesovec, Jan 05 2013

a(n) = Sum_{k = 0..floor((n-1)/2)} (n-2*k-1)!*binomial(n-k-1,k)* binomial(n-k+1,k+2). Cf. A058798. - Peter Bala, Aug 01 2013

a(n) = (n+1)!*hypergeometric([1/2-n/2,1-n/2],[3,-1-n,1-n],4)/2 for n >= 2. - Peter Luschny, Sep 10 2014

E.g.f.: 2*(I(2,2)*K(2, 2*sqrt(1-x)) - K(2,2)*I(2, 2*sqrt(1-x)))/(1-x), where I(n, x) and K(n, x) are the modified Bessel functions of the second kind. - G. C. Greubel, Oct 07 2019