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

0, 1, 5, 26, 157, 1100, 8801, 79210, 792101, 8713112, 104557345, 1359245486, 19029436805, 285441552076, 4567064833217, 77640102164690, 1397521838964421, 26552914940324000, 531058298806480001, 11152224274936080022

Offset: 0

Author

Olivier Gérard, Nov 02 2012

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000522, A002627, A056542.

Programs

Magma

[n le 1 select 0 else (n+1) * Self(n-1) + 1: n in [1..20]]; // Vincenzo Librandi, Dec 22 2012

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 *)

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! + ....

1/(e - 5/2) = 3! - 3!/(1*5) - 4!/(5*26) - 5!/(26*157) - 6!/(157*1100) - .... (see A002627, A056542). (End)

Extensions

Edited by Vincenzo Librandi and N. J. A. Sloane, Dec 24 2012