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A185108 a(0)=0; for n>0, a(n) = (n+2)*a(n-1) + 1.

%I A185108 #31 Aug 03 2023 16:04:49
%S A185108 0,1,5,26,157,1100,8801,79210,792101,8713112,104557345,1359245486,
%T A185108 19029436805,285441552076,4567064833217,77640102164690,
%U A185108 1397521838964421,26552914940324000,531058298806480001,11152224274936080022
%N A185108 a(0)=0; for n>0, a(n) = (n+2)*a(n-1) + 1.
%H A185108 Vincenzo Librandi, <a href="/A185108/b185108.txt">Table of n, a(n) for n = 0..200</a>
%F A185108 a(n) = e*Gamma(n+3,1)-(5/2)*(n+2)!, where Gamma(a,x) is the incomplete gamma function. [_Bruno Berselli_, Dec 24 2012]
%F A185108 Recurrence: a(n) = (n+3)*a(n-1) - (n+1)*a(n-2). - _Vaclav Kotesovec_, Aug 13 2013
%F A185108 a(n) ~ (exp(1)-5/2)*sqrt(2*Pi)*exp(-n)*n^(n+5/2). - _Vaclav Kotesovec_, Aug 13 2013
%F A185108 From _Peter Bala_, Oct 09 2013: (Start)
%F A185108 a(n) = A000522(n+2) - 5/2*(n + 2)! = (n + 2)!*( (sum {k = 0..n + 2} 1/k!) - 5/2 ).
%F A185108 a(n) = floor((n + 2)!*(e - 5/2)).
%F A185108 E.g.f.: ((x^2 - 4*x + 5)*exp(x) - 5)/(1 - x)^3 = x + 5*x^2/2! + 26*x^3/3! + ....
%F A185108 1/(e - 5/2) = 3! - 3!/(1*5) - 4!/(5*26) - 5!/(26*157) - 6!/(157*1100) - .... (see A002627, A056542). (End)
%t A185108 RecurrenceTable[{a[0]==0, a[n]==(n+2)*a[n-1] + 1}, a, {n, 20}] (* _Vincenzo Librandi_, Dec 23 2012 *)
%t A185108 nxt[{n_,a_}]:={n+1,a(n+3)+1}; NestList[nxt,{0,0},20][[;;,2]] (* _Harvey P. Dale_, Aug 03 2023 *)
%o A185108 (Magma) [n le 1 select 0 else (n+1) * Self(n-1) + 1: n in [1..20]]; // _Vincenzo Librandi_, Dec 22 2012
%Y A185108 Cf. A000522, A002627, A056542.
%K A185108 nonn
%O A185108 0,3
%A A185108 _Olivier Gérard_, Nov 02 2012
%E A185108 Edited by _Vincenzo Librandi_ and _N. J. A. Sloane_, Dec 24 2012