%I A006040 M1950
%S A006040 1,2,9,82,1313,32826,1181737,57905114,3705927297,300180111058,
%T A006040 30018011105801,3632179343801922,523033825507476769,88392716510763573962
%N A006040 a(n+1) = n^2 a(n) + 1.
%D A006040 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006040 R. K. Guy, personal communication.
%H A006040 <a href="Sindx_Be.html#Bessel">Index entries for sequences related to 
               Bessel functions or polynomials</a>
%F A006040 Nearest integer to BesselI(0, 2)*n!*n!, n>2.
%F A006040 a(n) = n!^2*Sum_{k=0..n} 1/k!^2. BesselI(0, 2*sqrt(x))/(1-x) = Sum_{n>
               =0} a(n)*x^n/n!^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 30 
               2002
%F A006040 Recurrence: a(1) = 1, a(2) = 2, a(n+1) = (n^2+1)*a(n) - (n-1)^2*a(n-1), 
               n >= 2. The sequence b(n) := (n-1)!^2 satisfies the same recurrence 
               with the initial conditions b(1) = 1, b(2) = 1. It follows that a(n) 
               = n!^2*(1 + 1/(1 - 1/(5 - 4/(10 - ...-(n-1)^2/(n^2+1))))). Hence 
               BesselI(0,2) := sum {k = 0..inf} 1/k!^2 = 1 + 1/(1 - 1/(5 - 4/(10 
               - ...-(n-1)^2/(n^2+1 - ...)))). Cf. A073701. - Peter Bala (pbala(AT)toucansurf.com), 
               Jul 09 2008
%t A006040 a = 1; lst = {a}; Do[a = a (n - 1)^2 + 1; AppendTo[lst, a], {n, 2, 14}]; 
               lst [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009]
%Y A006040 Main diagonal of array A099597.
%Y A006040 Cf. A073701.
%Y A006040 Sequence in context: A112670 A117581 A123570 this_sequence A067309 A087798 
               A113146
%Y A006040 Adjacent sequences: A006037 A006038 A006039 this_sequence A006041 A006042 
               A006043
%K A006040 nonn,easy
%O A006040 1,2
%A A006040 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)