OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
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]
Recurrence: a(n) = (n+3)*a(n-1) - (n+1)*a(n-2). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ (exp(1)-5/2)*sqrt(2*Pi)*exp(-n)*n^(n+5/2). - Vaclav Kotesovec, Aug 13 2013
From Peter Bala, Oct 09 2013: (Start)
a(n) = A000522(n+2) - 5/2*(n + 2)! = (n + 2)!*( (sum {k = 0..n + 2} 1/k!) - 5/2 ).
a(n) = floor((n + 2)!*(e - 5/2)).
E.g.f.: ((x^2 - 4*x + 5)*exp(x) - 5)/(1 - x)^3 = x + 5*x^2/2! + 26*x^3/3! + ....
MATHEMATICA
RecurrenceTable[{a[0]==0, a[n]==(n+2)*a[n-1] + 1}, a, {n, 20}] (* Vincenzo Librandi, Dec 23 2012 *)
nxt[{n_, a_}]:={n+1, a(n+3)+1}; NestList[nxt, {0, 0}, 20][[;; , 2]] (* Harvey P. Dale, Aug 03 2023 *)
PROG
(Magma) [n le 1 select 0 else (n+1) * Self(n-1) + 1: n in [1..20]]; // Vincenzo Librandi, Dec 22 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Olivier Gérard, Nov 02 2012
EXTENSIONS
Edited by Vincenzo Librandi and N. J. A. Sloane, Dec 24 2012
STATUS
approved