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

%I A145225 #29 Jan 31 2025 04:22:23
%S A145225 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,
%T A145225 945,420,210,0,21,0,0,7420,7392,3780,1120,420,0,28,0,0,66744,66780,
%U A145225 33264,11340,2520,756,0,36,0,0
%N A145225 Triangle read by rows: T(n,k) is the number of odd permutations (of an n-set) with exactly k fixed points.
%H A145225 Paolo Xausa, <a href="/A145225/b145225.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).
%H A145225 Bashir Ali and A. Umar, <a href="http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_200805&amp;filename=Some Combinatorial Properties of the Alternating Group.pdf">Some combinatorial properties of the alternating group</a>, Southeast Asian Bulletin Math. 32 (2008), 823-830.
%F A145225 T(n,k) = C(n,k) * A000387(n-k).
%F A145225 E.g.f.: x^(k+2) * exp(-x) / (2*k!*(1-x)).
%F A145225 T(n,k) + A145224(n,k) = A008290(n,k). - _R. J. Mathar_, Jul 06 2023
%F A145225 T(n,k) = (A008290(n,k) - A055137(n,k)) / 2. - _Julian Hatfield Iacoponi_, Aug 08 2024
%e A145225 Triangle starts:
%e A145225    0;
%e A145225    0,  0;
%e A145225    1,  0, 0;
%e A145225    0,  3, 0,  0;
%e A145225    6,  0, 6,  0, 0;
%e A145225   20, 30, 0, 10, 0, 0;
%e A145225   ...
%p A145225 A145225 := proc(n,k)
%p A145225     binomial(n,k)*A000387(n-k) ; # re-use code of A000387
%p A145225 end proc:
%p A145225 seq(seq(A145225(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Jul 06 2023
%t A145225 A145225[n_, k_] := Binomial[n, k]*Binomial[n - k, 2]*Subfactorial[n - k - 2];
%t A145225 Table[A145225[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Jan 31 2025 *)
%Y A145225 Row sums are A001710 for n > 1.
%Y A145225 Columns k=0..2 are A000387, A145222, A145223.
%Y A145225 Cf. A000387, A008290, A055137, A145224.
%K A145225 nonn,tabl
%O A145225 0,8
%A A145225 _Abdullahi Umar_, Oct 10 2008