A339350 Triangle read by rows. T(n,k) = Sum_{j=0..k} binomial(k-j+2, 2)*T(n-1, j), for n>=0, 0<=k<=n, with T(0,0)=1 and T(n,n)=0 for n>0.

1, 1, 0, 1, 3, 0, 1, 6, 15, 0, 1, 9, 39, 91, 0, 1, 12, 72, 272, 612, 0, 1, 15, 114, 570, 1995, 4389, 0, 1, 18, 165, 1012, 4554, 15180, 32890, 0, 1, 21, 225, 1625, 8775, 36855, 118755, 254475, 0, 1, 24, 294, 2436, 15225, 75516, 302064, 949344, 2017356, 0

Offset: 0

Author

Michel Marcus, Dec 01 2020

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1,  0;
  1,  3,  0;
  1,  6, 15,   0;
  1,  9, 39,  91,   0;
  1, 12, 72, 272, 612, 0;
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).
• Francesca Aicardi, Catalan triangle and tied arc diagrams, arXiv:2011.14628 [math.CO], 2020.
• Francesca Aicardi, Fuss-Catalan Triangles, arXiv:2310.07317 [math.CO], 2023.

Crossrefs

Cf. subdiagonals: A006632, A006633, A006634, A006635.

Cf. A002293 (row sums).

Programs

Mathematica

A339350[n_, k_] := A339350[n, k] = Which[k == 0, 1, n == k, 0, True, Sum[Binomial[k-j+2, 2]*A339350[n-1, j], {j, 0, k}]];
Table[A339350[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 23 2024 *)

PARI

T(n,k) = if ((n==0) && (k==0), 1, if (n<=k, 0, sum(j=0, k, binomial(k-j+2, 2)*T(n-1, j))));