A161134 - OEIS

%I #17 May 19 2021 03:39:11

%S 1,1,1,1,4,2,14,8,2,78,36,6,426,234,54,6,3216,1512,288,24,24024,12864,

%T 3024,384,24,229080,108960,22320,2400,120,2170680,1145400,272400,

%U 37200,3000,120,25022880,11998800,2563200,309600,21600,720,287250480

%N 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 Row n contains 1 + floor(n/2) entries.

%C Sum of row n is n! = A000142(n).

%C T(n,0) = A161132(n).

%C Sum_{k>=0} k*T(n,k) = A052591(n-1).

%H Indranil Ghosh, <a href="/A161134/b161134.txt">Rows 0..125, flattened</a>

%F 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 T(3,0)=4 because we have 132, 312, 213, 231; T(3,1)=2 because we have 123 and 321.

%e Triangle starts:

%e     1;

%e     1;

%e     1,   1;

%e     4,   2;

%e    14,   8,   2;

%e    78,  36,   6;

%e   426, 234,  54,   6;

%p 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 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 (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 tabf(12); \\ _Indranil Ghosh_, Mar 08 2017

%o (Python)

%o from sympy import factorial, binomial

%o def T(n,k):

%o     s=0

%o     for j in range(n//2 - k+1):

%o         s+=(-1)**j * factorial(n-k-j) * binomial(n//2 - k, j)

%o     return binomial(n//2, k)* s

%o i=0

%o for n in range(26):

%o     for k in range(n//2 + 1):

%o         print(str(i)+" "+str(T(n,k)))

%o         i+=1

%o # _Indranil Ghosh_, Mar 08 2017

%Y Cf. A000142, A052591, A161132, A161133.

%K nonn,tabf

%O 0,5

%A _Emeric Deutsch_, Jul 18 2009