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%I #22 Aug 25 2024 19:47:22
%S 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 945,420,210,0,21,0,0,7420,7392,3780,1120,420,0,28,0,0,66744,66780,
%U 33264,11340,2520,756,0,36,0,0
%N Triangle read by rows: T(n,k) is the number of odd permutations (of an n-set) with exactly k fixed points.
%H Bashir Ali and A. Umar, <a href="http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_200805&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 T(n,k) = C(n,k) * A000387(n-k).
%F E.g.f.: x^(k+2) * exp(-x) / (2*k!*(1-x)).
%F T(n,k) + A145224(n,k) = A008290(n,k). - _R. J. Mathar_, Jul 06 2023
%F T(n,k) = (A008290(n,k) - A055137(n,k)) / 2. - _Julian Hatfield Iacoponi_, Aug 08 2024
%e Triangle starts:
%e 0;
%e 0, 0;
%e 1, 0, 0;
%e 0, 3, 0, 0;
%e 6, 0, 6, 0, 0;
%e 20, 30, 0, 10, 0, 0;
%e ...
%p A145225 := proc(n,k)
%p binomial(n,k)*A000387(n-k) ; # re-use code of A000387
%p end proc:
%p seq(seq(A145225(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Jul 06 2023
%Y Row sums are A001710 for n > 1.
%Y Columns k=0..2 are A000387, A145222, A145223.
%Y Cf. A008290, A055137.
%K nonn,tabl,changed
%O 0,8
%A _Abdullahi Umar_, Oct 10 2008
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