A145224 Triangle read by rows: T(n,k) is the number of even permutations (of an n-set) with exactly k fixed points.

1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 3, 8, 0, 0, 1, 24, 15, 20, 0, 0, 1, 130, 144, 45, 40, 0, 0, 1, 930, 910, 504, 105, 70, 0, 0, 1, 7413, 7440, 3640, 1344, 210, 112, 0, 0, 1, 66752, 66717, 33480, 10920, 3024, 378, 168, 0, 0, 1

Offset

0,7

Links

Table of n, a(n) for n=0..54.

Bashir Ali and A. Umar, Some combinatorial properties of the alternating group, Southeast Asian Bulletin Math. 32 (2008), 823-830.

Formula

T(n,k) = C(n,k)*A003221(n-k).

E.g.f.: (x^k(1-x^2/2) e^(-x))/k!(1-x).

T(n,k) + A145225(n,k) = A008290(n,k). - R. J. Mathar, Jul 06 2023

T(n,k) = (A008290(n,k) + A055137(n,k))/2. - Julian Hatfield Iacoponi, Aug 08 2024

Example

Triangle starts:
   1;
   0,  1;
   0,  0,  1;
   2,  0,  0, 1;
   3,  8,  0, 0, 1;
  24, 15, 20, 0, 0, 1;
  ...

Crossrefs

Row sums give A001710.

Columns k=0..2 are A003221, A145219, A145220.

Cf. A008290, A055137.

Sequence in context: A373183 A351776 A259784 * A138157 A342243 A073429

Adjacent sequences: A145221 A145222 A145223 * A145225 A145226 A145227

Keyword

nonn,tabl

Author

Abdullahi Umar, Oct 09 2008

Status

approved