|
%I #29 Dec 23 2023 13:09:08
%S 1,2,3,8,27,116,607,3758,26913,219062,1998471,20203772,224239963,
%T 2711083328,35468323227,499267608506,7524482450817,120890986821578,
%U 2062671258417643,37248973638339152,709793170386861531
%N 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).
%H Vincenzo Librandi, <a href="/A093858/b093858.txt">Table of n, a(n) for n = 0..200</a>
%F 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
%F 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
%F Lim_{n->infinity} a(n)/(n-1)! = 2*BesselI(0,2) - BesselI(1,-2) = 6.1498074593094635982566633... - _Vaclav Kotesovec_, Jan 05 2013
%t 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 *)
%t nxt[{n_,a_,b_}]:={n+1,b,b*n+a}; NestList[nxt,{1,1,2},20][[;;,2]] (* _Harvey P. Dale_, Dec 23 2023 *)
%Y Similar recurrences: A001040, A001053, A058279, A058307. - _Wolfdieter Lang_, May 19 2010
%K easy,nonn
%O 0,2
%A _Amarnath Murthy_, Apr 19 2004
%E a(10)-a(20) from _Ryan Propper_, Sep 14 2005
|
