A145881 Triangle read by rows: T(n,k) is the number of even permutations of {1,2,...,n} with no fixed points and having k excedances (n>=1; k>=1).

0, 0, 1, 1, 0, 3, 0, 1, 11, 11, 1, 0, 25, 80, 25, 0, 1, 57, 407, 407, 57, 1, 0, 119, 1680, 3815, 1680, 119, 0, 1, 247, 6211, 26917, 26917, 6211, 247, 1, 0, 501, 21432, 160053, 303504, 160053, 21432, 501, 0, 1, 1013, 70775, 852347, 2747009, 2747009, 852347, 70775

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 = A000321(n).

Sum_{k=1..n-1} k*T(n,k) = A145887(n) (n>=2).

Examples

			T(4,2)=3 because the even derangements of {1,2,3,4} are 3412, 2143 and 4321.
Triangle starts:
  0;
  0;
  1,  1;
  0,  3,  0;
  1, 11, 11,  1;
  0, 25, 80, 25,  0;
		

Links

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

Crossrefs

Cf. A000321, A145880, A145886, 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 Jon E. Schoenfield, Jul 21 2017 at the request of the author