A050157 - OEIS

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

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OFFSET

0,3

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)}.

LINKS

Table of n, a(n) for n=0..44.

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

EXAMPLE

The triangle starts:

1, 2

2, 5, 6

5, 14, 19, 20

14, 42, 62, 69, 70

42, 132, 207, 242, 251, 252