1,2

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, personal communication.

Index entries for sequences related to Bessel functions or polynomials

Nearest integer to BesselI(0, 2)*n!*n!, n>2.

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

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

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]

Main diagonal of array A099597.

Cf. A073701.

Sequence in context: A112670 A117581 A123570 this_sequence A067309 A087798 A113146

Adjacent sequences: A006037 A006038 A006039 this_sequence A006041 A006042 A006043

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)