A172371 Antidiagonal triangle T(n, k) = t(k, n-k+1) where the square array t(n,k) is defined by t(n, k) = k*t(n-2, k) + t(n-3, k), t(0, k) = 0, and t(1, k) = t(2, k) = 1, read by rows.

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 3, 3, 2, 0, 1, 1, 4, 4, 5, 3, 0, 1, 1, 5, 5, 10, 8, 4, 0, 1, 1, 6, 6, 17, 15, 13, 5, 0, 1, 1, 7, 7, 26, 24, 34, 21, 7, 0, 1, 1, 8, 8, 37, 35, 73, 55, 34, 9, 0, 1, 1, 9, 9, 50, 48, 136, 113, 117, 55, 12

Offset

0,14

Links

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Formula

T(n, k) = t(k, n-k+1) where the square array t(n,k) is defined by t(n, k) = k*t(n-2, k) + t(n-3, k), t(0, k) = 0, and t(1, k) = t(2, k) = 1.

Example

The square array t(n, k) begins as:
  0, 0, 0,  0,  0,  0, ...
  1, 1, 1,  1,  1,  1, ...
  1, 1, 1,  1,  1,  1, ...
  0, 1, 2,  3,  4,  5, ...
  1, 2, 3,  4,  5,  6, ...
  1, 2, 5, 10, 17, 26, ...
Antidiagonal triangle begins as:
  0;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 1, 2, 2;
  0, 1, 1, 3, 3,  2;
  0, 1, 1, 4, 4,  5,  3;
  0, 1, 1, 5, 5, 10,  8,  4;
  0, 1, 1, 6, 6, 17, 15, 13,  5;
  0, 1, 1, 7, 7, 26, 24, 34, 21, 7;

Mathematica

T[n_, k_]:= T[n, k]= If[n==0, 0, If[n<3, 1, k*T[n-2, k] +T[n-3, k] ]];
Table[T[k, n-k+1], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 02 2021 *)

Prog

@CachedFunction
def T(n, k):
    if (n==0): return 0
    elif (n<3): return 1
    else: return k*T(n-2, k) + T(n-3, k)
flatten([[T(k, n-k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 02 2021

Crossrefs

Sequence in context: A306910 A112185 A192062 * A279006 A112555 A108561

Adjacent sequences: A172368 A172369 A172370 * A172372 A172373 A172374

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 01 2010

Extensions

Edited by G. C. Greubel, May 02 2021

Status

approved