1,6

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

T(n,k) is the number of derangements of [n] having k excedances. Example: T(4,2)=7 because we have 3*14*2, 3*4*12, 4*3*12, 2*14*3, 2*4*13, 3*4*21, 4*3*21, each with two excedances (marked). An excedance of a permutation p is a position i such that p(i)>i.

Sum(k*T(n,k),k>=1)=A000274(n+1).

(End)

D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16.

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, r)=r*a(n, r)+(n+1-r)a(n, r-1)+n*a(n-1, r-1).

exp(-t)/(1 - exp((x-1)t)/(x-1)) = 1 + x*t^2/2! + (x+x^2)*t^3/3! + (x+7x^2+x^3)*t^4/4! + (x+21x^2+21x^3+x^4)*t^5/5! + ... - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 11 2004

0; 1; 1 1; 1 7 1; 1 21 21 1; 1 51 161 51 1; ...

G := (1-t)*exp(-t*z)/(1-t*exp((1-t)*z)): Gser := simplify(series(G, z = 0, 15)): for n to 13 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: 0; for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n-1) end do; # yields sequence in triangular form [From Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2009]

Cf. A046740. Row sums give A000166. Diagonals give A070313, A070315.

A000274 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2009]

Sequence in context: A154233 A119727 A157272 this_sequence A056752 A053714 A142465

Adjacent sequences: A046736 A046737 A046738 this_sequence A046740 A046741 A046742

nonn,easy,nice,tabf

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000