OFFSET
0,6
COMMENTS
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
T(2n+1,k) = n*k!*(2n-k)!*binomial(n+1,k) (n>= 1);
T(2n,k) = n*k!*(2n-1-k)!*binomial(n,k).
From Alois P. Heinz, Apr 02 2024: (Start)
Sum_{k>=0} (k+1) * T(n,k) = A256881(n+1).
T(n,ceiling(n/2)) = A010551(n). (End)
EXAMPLE
T(3,0)=2 because we have 213 and 231.
T(4,2)=4 because we have 1324, 1342, 3124 and 3142.
Triangle starts:
1;
0, 1;
1, 1;
2, 2, 2;
12, 8, 4;
48, 36, 24, 12;
360, 216, 108, 36;
...
MAPLE
T := proc (n, k) if n=0 and k=0 then 1 elif n = 1 and k = 0 then 0 elif n = 1 and k = 1 then 1 elif `mod`(n, 2) = 1 then (1/2)*(n-1)*binomial((1/2)*n+1/2, k)*factorial(k)*factorial(n-1-k) else (1/2)*n*binomial((1/2)*n, k)*factorial(k)*factorial(n-1-k) end if end proc: for n from 0 to 11 do seq(T(n, k), k = 0 .. ceil((1/2)*n)) end do; # yields sequence in triangular form
MATHEMATICA
T[n_, k_] := T[n, k] = Which[n == 0 && k == 0, 1, n == 1 && k == 1, 1, OddQ[n], (n - 1)/2*k!*(n - k - 1)!*Binomial[(n - 1)/2 + 1, k], True, n/2*k!*(n - k - 1)!*Binomial[n/2, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, Ceiling[n/2]}] // Flatten (* Jean-François Alcover, Apr 04 2024 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Dec 13 2008
EXTENSIONS
T(0,0)=1 prepended by Alois P. Heinz, Apr 02 2024
STATUS
approved