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

0,2

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A008517.

Sequence in context: A247943 A329571 A270358 * A184712 A303225 A326943

Adjacent sequences: A156526 A156527 A156528 * A156530 A156531 A156532

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 09 2009

Extensions

Edited by G. C. Greubel, Dec 30 2021

Status

approved