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

Author

Emeric Deutsch, Nov 06 2008

Keywords

• nonn
• tabf

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).

Examples

			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;
		

Links

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

Crossrefs

Cf. A000387, A145886, A145881, A145887.

Programs

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

Formula

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

Extensions

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