A156529 Triangle, T(n, k) = A008517(n+1, k+1)*A008517(n+1, n-k+1), read by rows.

1, 2, 2, 6, 64, 6, 24, 1276, 1276, 24, 120, 23088, 107584, 23088, 120, 720, 422712, 6388800, 6388800, 422712, 720, 5040, 8156160, 326165400, 1031694400, 326165400, 8156160, 5040, 40320, 168521184, 15666814800, 126099116000, 126099116000, 15666814800, 168521184, 40320

Offset: 0

Author

Roger L. Bagula, Feb 09 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
     1;
     2,       2;
     6,      64,         6;
    24,    1276,      1276,         24;
   120,   23088,    107584,      23088,       120;
   720,  422712,   6388800,    6388800,    422712,     720;
  5040, 8156160, 326165400, 1031694400, 326165400, 8156160, 5040;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A008517.

Programs

Magma

A008517:= func< n,k | (&+[ (-1)^(n+j)*Binomial(2*n+1, j)*StiringFirst(2*n-k-j+1, n-k-j+1) : j in [0..n-k]]) >;
A156529:= func< n,k | A008517(n+1,k+1)*A008517(n+1,n-k+1) >;
[A156529(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 30 2021

Mathematica

f[n_, k_]:= f[n, k]= If[k<0 || k>n, 0, If[k==0, 1, (k+1)*f[n-1, k] + (2*n-k+1)*f[n-1, k-1] ]]; (* f = A008517 *)
T[n_, k_]:= f[n+1, k+1]*f[n+1, n-k+1];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Dec 30 2021 *)

Sage

@CachedFunction
def A008517(n,k): return sum( (-1)^(n+j)*binomial(2*n+1, j)*stirling_number1(2*n-k-j+1, n-k-j+1)  for j in (0..n-k) )
def A156529(n,k): return A008517(n+1, k+1)*A008517(n+1, n-k+1)
flatten([[A156529(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 30 2021

Formula

T(n, k) = A008517(n+1, k+1)*A008517(n+1, n-k+1).

From G. C. Greubel, Dec 30 2021: (Start)

T(n, n-k) = T(n, k).

T(n, 0) = n!. (End)

Extensions

Edited by G. C. Greubel, Dec 30 2021