A050157 T(n, k) = S(2n, n, k) for 0<=k<=n and n>=0, where S(p, q, r) is the number of upright paths from (0, 0) to (p, p-q) that do not rise above the line y = x-r.

1, 1, 2, 2, 5, 6, 5, 14, 19, 20, 14, 42, 62, 69, 70, 42, 132, 207, 242, 251, 252, 132, 429, 704, 858, 912, 923, 924, 429, 1430, 2431, 3068, 3341, 3418, 3431, 3432, 1430, 4862, 8502, 11050, 12310, 12750, 12854, 12869, 12870

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

Let V = (e(1),...,e(n)) consist of q 1's and p-q 0's; let V(h) = (e(1),...,e(h)) and m(h) = (#1's in V(h)) - (#0's in V(h)) for h=1,...,n. Then S(p,q,r) is the number of V having r >= max{m(h)}.

Examples

			The triangle starts:
                                1
                              1, 2
                            2, 5, 6
                         5, 14, 19, 20
                       14, 42, 62, 69, 70
                  42, 132, 207, 242, 251, 252
               132, 429, 704, 858, 912, 923, 924
		

Crossrefs

T(n, 0) = A000108(n).

T(n, 1) = A000108(n+1).

T(n, n) = A000984(n).

T(n, n-1) = A030662(n).

Row sums are A296771.

Cf. A039599, A050163.

Programs

Maple

A050157 := (n, k) -> binomial(2*n, n) - binomial(2*n, n+k+1):
seq(seq(A050157(n,k), k=0..n), n=0..10); # Peter Luschny, Dec 21 2017

Formula

T(n, k) = Sum_{0<=j<=k} t(n, j), array t as in A039599.

T(n, k) = binomial(2*n, n) - binomial(2*n, n+k+1). - Peter Luschny, Dec 21 2017