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a(n) = n^2 * a(n-1) + n, a(0)=0.
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%I #24 Jan 05 2024 07:57:26

%S 0,1,6,57,916,22905,824586,40404721,2585902152,209458074321,

%T 20945807432110,2534442699285321,364959748697086236,

%U 61678197529807573897,12088926715842284483826,2720008511064514008860865

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

%C 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).

%H Vincenzo Librandi, <a href="/A180255/b180255.txt">Table of n, a(n) for n = 0..41</a>

%F From _Seiichi Manyama_, Jan 05 2024: (Start)

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

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

%o (PARI)

%o a(n)=if(n==0,0,(n)^2*a(n-1)+(n));

%o for(n=0,12,print1(a(n),", ")); /* show terms */

%o (Maxima) a[0]:0$ a[n]:=n^2*a[n-1]+n$ makelist(a[n], n, 0, 15); /* _Bruno Berselli_, May 23 2011 */

%Y Cf. A006040, A007526, A066998, A228229.

%K nonn

%O 0,3

%A _Groux Roland_, Jan 17 2011