OFFSET
0,3
LINKS
W. Y. C. Cheng, E. Y. P. Deng, R. R. X. Du, R. P. Stanley and C. H. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.
FORMULA
T(n, k) = Sum_{i>=0} (binomial(2n, n-i*(k+2)) - binomial(2n, n+i*(k+2)+k+1)).
EXAMPLE
Triangle begins
1;
1, 2;
3, 5, 6;
10, 15, 19, 20;
35, 50, 63, 69, 70;
126, 176, 217, 243, 251, 252;
462, 638, 770, 870, 913, 923, 924;
1716, 2354, 2794, 3159, 3355, 3419, 3431, 3432;
6435, 8789, 10307, 11610, 12430, 12766, 12855, 12869, 12870;
MATHEMATICA
T[n_, k_] := Sum[Binomial[2n, n-i(k+2)] - Binomial[2n, n+i(k+2)+k+1], {i, 0, n}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 20 2019 *)
CROSSREFS
KEYWORD
AUTHOR
Ralf Stephan, Jan 21 2005
STATUS
approved