A180255 a(n) = n^2 * a(n-1) + n, a(0)=0.

0, 1, 6, 57, 916, 22905, 824586, 40404721, 2585902152, 209458074321, 20945807432110, 2534442699285321, 364959748697086236, 61678197529807573897, 12088926715842284483826, 2720008511064514008860865, 696322178832515586268381456, 201237109682597004431562240801

Offset: 0

Author

Groux Roland, Jan 17 2011

Keywords

• nonn

Comments

Integral_{x=0..1} x^n*BesselI(0,2*x^(1/2)) dx = A006040(n)*BesselI(1,2) - a(n)*BesselI(0,2). An elementary consequence is the irrationality of BesselI(0,2)/BesselI(1,2).

Links

• Paolo Xausa, Table of n, a(n) for n = 0..250 (terms 0..41 from Vincenzo Librandi)

Crossrefs

Cf. A006040, A007526, A066998, A228229.

Programs

Mathematica

FoldList[#2^2*# + #2 &, Range[0, 20]] (* Paolo Xausa, Jun 19 2025 *)

Maxima

a[0]:0$ a[n]:=n^2*a[n-1]+n$ makelist(a[n], n, 0, 15); /* Bruno Berselli, May 23 2011 */

PARI

a(n)=if(n==0,0,(n)^2*a(n-1)+(n));
for(n=0,12,print1(a(n),", "));  /* show terms */

Formula

From Seiichi Manyama, Jan 05 2024: (Start)

a(n) = (n!)^2 * Sum_{k=0..n} k/(k!)^2.

a(n) = n * A228229(n-1) for n > 0. (End)