A066998 a(0)=0; a(n) = n^2*a(n-1) + 1.

0, 1, 5, 46, 737, 18426, 663337, 32503514, 2080224897, 168498216658, 16849821665801, 2038828421561922, 293591292704916769, 49616928467130933962, 9724917979557663056553, 2188106545400474187724426

Offset: 0

Author

Benoit Cloitre, Jan 27 2002

Keywords

• nonn

Comments

if s(n) is a sequence defined as s(0)=x, s(n) = n^2*s(n-1) + k, then s(n) = n!^2*x + a(n)*k. - Gary Detlefs, Feb 20 2010

Links

• Harry J. Smith, Table of n, a(n) for n = 0..100
• Alexandre Silva, Carlos Pereira dos Santos, João Pedro Neto, and Richard J. Nowakowski, Disjunctive sums of quasi-nimbers, Theor. Comp. Sci. (2022).

Crossrefs

This is the same recurrence relation as A006040 except A006040 has a(0) = 1.

Programs

Mathematica

RecurrenceTable[{a[0]==0,a[n]==n^2 a[n-1]+1},a,{n,20}] (* Harvey P. Dale, Jan 24 2019 *)

PARI

a(n) = { my(a=0); for (n=1, n, a=n^2*a + 1); a } \\ Harry J. Smith, Apr 24 2010

Formula

a(n) = (n!)^2*Sum_{i=1..n} 1/(i!)^2.

a(n) = floor((1-BesselI(0, 2))*(n!)^2). - Benoit Cloitre, Nov 02 2002

Sum_{n>=0} a(n) * x^n / (n!)^2 = (BesselI(0,2*sqrt(x)) - 1) / (1 - x). - Ilya Gutkovskiy, Jan 23 2021

Extensions

Better description from James D. Klein, Feb 25 2002