A145225 T(n,k) is the number of odd permutations (of an n-set) with exactly k fixed points.

0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 6, 0, 6, 0, 0, 20, 30, 0, 10, 0, 0, 135, 120, 90, 0, 15, 0, 0, 924, 945, 420, 210, 0, 21, 0, 0, 7420, 7392, 3780, 1120, 420, 0, 28, 0, 0, 66744, 66780, 33264, 11340, 2520, 756, 0, 36, 0, 0

Offset

0,8

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)*A000387(n-k)

E.g.f.: (x^(k+2)*exp(-x))/(2*(k!)*(1-x)).

T(n,k) + A145224(n,k) = A008290(n,k). - R. J. Mathar, Jul 06 2023

Example

Triangle starts:
0;
0, 0;
1, 0, 0;
0, 3, 0, 0;
6, 0, 6, 0, 0;
20, 30, 0, 10, 0;
...

Maple

A145225 := proc(n, k)
    binomial(n, k)*A000387(n-k) ; # re-use code of A000387
end proc:
seq(seq(A145225(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Jul 06 2023

Crossrefs

Row sum is A001710 for n > 1, sum of Row1=sum of Row2 = 0.

T(n, 0) is A000387, T(n, 1) is A145222, T(n, 2) is A145223.

Sequence in context: A062688 A067181 A321429 * A332442 A061480 A220692

Adjacent sequences: A145222 A145223 A145224 * A145226 A145227 A145228

Keyword

nonn,tabl

Author

Abdullahi Umar, Oct 10 2008

Status

approved