3,2

Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2009: (Start)

a(n)=number of excedances in all derangements of [n-1]. Example: a(5)=18 because the derangements of {1,2,3,4} are 4*123, 3*14*2, 3*4*12, 4*3*12, 2*14*3, 2*4*13, 2*3*4*1, 3*4*21, 4*3*21 with the 18 excedances marked. An excedance of a permutation p is a position i such that p(i)>i.

a(n)=Sum(k*A046739(n,k), k>=1).

(End)

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210 (divided by 2).

R. Mantaci and F. Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2009]

a(n) = (1 + n) a(n - 1) + (3 + n) a(n - 2) + (3 - n) a(n - 3) + (2 - n) a(n - 4).

E.g.f.: x^2/2*exp(-x)/(1-x)^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 03 2003

a(n)=(n-1)^2/(n-2)*a(n-1)-(-1)^n*(n-1)/2, n>2, a(2)=0. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 31 2003

(1/2){[n!/e] - [(n-1)!/e]} (conjectured).

a(n) = (n-1)*GAMMA(n,-1)*exp(-1)/2 where GAMMA = incomplete Gamma function [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 11 2009]

a:=n->sum(n!*sum((-1)^k/k!/2, j=1..n), k=0..n): seq(a(n), n=2..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2007

Table[Subfactorial[n]*n/2, {n, 2, 20}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2009]

Cf. A010027, A000255, A000166, A000313, A001260, A001261.

A diagonal in triangle A010027.

A046739 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2009]

Sequence in context: A074571 A114311 A134092 this_sequence A163471 A054122 A074566

Adjacent sequences: A000271 A000272 A000273 this_sequence A000275 A000276 A000277

easy,nonn,new

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)