A093858 a(0) = 1, a(1)= 2, a(n) = (a(n+1) - a(n-1))/n, or a(n+1) = n*a(n) + a(n-1).

1, 2, 3, 8, 27, 116, 607, 3758, 26913, 219062, 1998471, 20203772, 224239963, 2711083328, 35468323227, 499267608506, 7524482450817, 120890986821578, 2062671258417643, 37248973638339152, 709793170386861531

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = -2*(BesselI[n, -2]*(2 BesselK[0, 2] - BesselK[1, 2]) + (-2 BesselI[0, 2] + BesselI[1, -2])*BesselK[n, 2]). - Ryan Propper, Sep 14 2005

E.g.f.: -3*Pi*(BesselI(1, 2)*BesselY(0, 2*I*sqrt(1-x)) + I*BesselY(1, 2*I)*BesselI(0, 2*sqrt(1-x))). Such e.g.f. computations were the result of an e-mail exchange with Gary Detlefs. After differentiation and setting x=0 one has to use simplifications. See the Abramowitz-Stegun handbook, p. 360, 9.1.16 and p. 375, 9.63. - Wolfdieter Lang, May 19 2010

Lim_{n->infinity} a(n)/(n-1)! = 2*BesselI(0,2) - BesselI(1,-2) = 6.1498074593094635982566633... - Vaclav Kotesovec, Jan 05 2013

Mathematica

a = 1; b = 2; Print[a]; Print[b]; Do[c = n*b + a; Print[c]; a = b; b = c, {n, 1, 30}] (* Ryan Propper, Sep 14 2005 *)
nxt[{n_, a_, b_}]:={n+1, b, b*n+a}; NestList[nxt, {1, 1, 2}, 20][[;; , 2]] (* Harvey P. Dale, Dec 23 2023 *)

Crossrefs

Similar recurrences: A001040, A001053, A058279, A058307. - Wolfdieter Lang, May 19 2010

Sequence in context: A086613 A121401 A318895 * A080568 A091339 A006277

Adjacent sequences: A093855 A093856 A093857 * A093859 A093860 A093861

Keyword

easy,nonn

Author

Amarnath Murthy, Apr 19 2004

Extensions

a(10)-a(20) from Ryan Propper, Sep 14 2005

Status

approved