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A006348 a(n) = (n+2)*a(n-1) + (-1)^n.
(Formerly M3609)
1

%I M3609

%S 0,1,4,25,174,1393,12536,125361,1378970,16547641,215119332,3011670649,

%T 45175059734,722800955745,12287616247664,221177092457953,

%U 4202364756701106,84047295134022121,1764993197814464540,38829850351918219881,893086558094119057262

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

%C a(n) is a function of the subfactorials... a(n) = A000166(n+2) - 1/3*(n+2)! /Q, i.e., ... 1 = 9 - 24/3, 4 = 44 - 120/3, 25 = 265 - 720/3 ... - _Gary Detlefs_, Dec 17 2009

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Robert Israel, <a href="/A006348/b006348.txt">Table of n, a(n) for n = 1..448</a>

%H J. A. Sharp & N. J. A. Sloane, <a href="/A002026/a002026.pdf">Correspondence, 1977</a>

%F a(n) = (n+1)(a(n-1) + a(n-2)). - _Gary Detlefs_, Dec 17 2009

%F E.g.f. with offset 0: ((2 + 3*x + x^3)*exp(-x) - 2)/(1 - x)^4. From int(((9 + 8*x + 6*x^2 + x^4)*exp(-x) - 8)/(1 - x)^5, x) with input 0 for x = 0. - _Wolfdieter Lang_, May 03 2010

%F From _Robert Israel_, Feb 28 2017: (Start)

%F a(n) = Gamma(n+3, -1)/e - (n+2)!/3.

%F a(n) ~ (1/e - 1/3) sqrt(2 Pi) n^(n+5/2) exp(-n). (End)

%p a:= n-> (n+2)!*sum((-1)^k/k!, k=4..n+2): seq(a(n), n=1..23); # _Zerinvary Lajos_, May 25 2007

%p a:= n-> floor(((n+2)!+1)/exp(1)) -(n+2)!/3: seq(a(n), n=1..23); # _Gary Detlefs_, Dec 17 2009

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_

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Last modified June 20 23:24 EDT 2021. Contains 345312 sequences. (Running on oeis4.)