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A161134 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having exactly k even fixed points (0 <= k <= floor(n/2)).

%I A161134 #17 May 19 2021 03:39:11
%S A161134 1,1,1,1,4,2,14,8,2,78,36,6,426,234,54,6,3216,1512,288,24,24024,12864,
%T A161134 3024,384,24,229080,108960,22320,2400,120,2170680,1145400,272400,
%U A161134 37200,3000,120,25022880,11998800,2563200,309600,21600,720,287250480
%N A161134 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having exactly k even fixed points (0 <= k <= floor(n/2)).
%C A161134 Row n contains 1 + floor(n/2) entries.
%C A161134 Sum of row n is n! = A000142(n).
%C A161134 T(n,0) = A161132(n).
%C A161134 Sum_{k>=0} k*T(n,k) = A052591(n-1).
%H A161134 Indranil Ghosh, <a href="/A161134/b161134.txt">Rows 0..125, flattened</a>
%F A161134 T(n,k) = binomial(floor(n/2), k)*Sum_{j=0..floor(n/2)-k}(-1)^j*(n-k-j)!*binomial(floor(n/2)-k, j).
%e A161134 T(3,0)=4 because we have 132, 312, 213, 231; T(3,1)=2 because we have 123 and 321.
%e A161134 Triangle starts:
%e A161134     1;
%e A161134     1;
%e A161134     1,   1;
%e A161134     4,   2;
%e A161134    14,   8,   2;
%e A161134    78,  36,   6;
%e A161134   426, 234,  54,   6;
%p A161134 T := proc (n, k) options operator, arrow: binomial(floor((1/2)*n), k)*add((-1)^j*binomial(floor((1/2)*n)-k, j)*factorial(n-k-j), j = 0 .. floor((1/2)*n)-k) end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%t A161134 Flatten[Table[Binomial[Floor[n/2], k]*Sum[(-1)^j*(n - k - j)!*Binomial[Floor[n/2] - k, j], {j, 0, Floor[n/2] - k}],{n, 0, 12}, {k, 0, Floor[n/2]}]] (* _Indranil Ghosh_, Mar 08 2017 *)
%o A161134 (PARI) tabf(nn) = { for(n=0, nn, for(k = 0, floor(n/2), print1(binomial(floor(n/2), k) * sum(j=0, floor(n/2) - k, (-1)^j*(n - k - j)! * binomial(floor(n/2) - k, j)),", ");); print();); };
%o A161134 tabf(12); \\ _Indranil Ghosh_, Mar 08 2017
%o A161134 (Python)
%o A161134 from sympy import factorial, binomial
%o A161134 def T(n,k):
%o A161134     s=0
%o A161134     for j in range(n//2 - k+1):
%o A161134         s+=(-1)**j * factorial(n-k-j) * binomial(n//2 - k, j)
%o A161134     return binomial(n//2, k)* s
%o A161134 i=0
%o A161134 for n in range(26):
%o A161134     for k in range(n//2 + 1):
%o A161134         print(str(i)+" "+str(T(n,k)))
%o A161134         i+=1
%o A161134 # _Indranil Ghosh_, Mar 08 2017
%Y A161134 Cf. A000142, A052591, A161132, A161133.
%K A161134 nonn,tabf
%O A161134 0,5
%A A161134 _Emeric Deutsch_, Jul 18 2009