A145880 Triangle read by rows: T(n,k) is the number of odd permutations of {1,2,...,n} with no fixed points and having k excedances (n>=1; k>=1).

0, 1, 0, 0, 1, 4, 1, 0, 10, 10, 0, 1, 26, 81, 26, 1, 0, 56, 406, 406, 56, 0, 1, 120, 1681, 3816, 1681, 120, 1, 0, 246, 6210, 26916, 26916, 6210, 246, 0, 1, 502, 21433, 160054, 303505, 160054, 21433, 502, 1, 0, 1012, 70774, 852346, 2747008, 2747008, 852346, 70774

Offset

1,6

Comments

Row n has n-1 entries (n>=2).

Sum of entries in row n = A000387(n).

Sum_{k=1..n-1} k*T(n,k) = A145886(n) (n>=2).

Links

Table of n, a(n) for n=1..54.

R. Mantaci and F. Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188.

Formula

E.g.f.: ((1-t)*exp(-tz)/(1-t*exp((1-t)z)) + (t*exp(-z)-exp(-tz))/(1-t))/2.

Example

T(4,2)=4 because the odd derangements of {1,2,3,4} with 2 excedances are 3142, 4312, 2413 and 3421.
Triangle starts:
  0;
  1;
  0,  0;
  1,  4,  1;
  0, 10, 10,  0;
  1, 26, 81, 26,  1;

Maple

G:=((1-t)*exp(-t*z)/(1-t*exp((1-t)*z))+(t*exp(-z)-exp(-t*z))/(1-t))*1/2: Gser:=simplify(series(G, z=0, 15)): for n to 11 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

Crossrefs

Cf. A000387, A145886, A145881, A145887.

Sequence in context: A363971 A127155 A211793 * A048516 A060638 A244125

Adjacent sequences: A145877 A145878 A145879 * A145881 A145882 A145883

Keyword

nonn,tabf

Author

Emeric Deutsch, Nov 06 2008

Extensions

Formula corrected by N. J. A. Sloane, Jul 20 2017 at the request of the author.

Status

approved