A050157 - OEIS

%I #12 Dec 21 2017 07:48:14

%S 1,1,2,2,5,6,5,14,19,20,14,42,62,69,70,42,132,207,242,251,252,132,429,

%T 704,858,912,923,924,429,1430,2431,3068,3341,3418,3431,3432,1430,4862,

%U 8502,11050,12310,12750,12854,12869,12870

%N 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.

%C 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)}.

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

%F T(n, k) = binomial(2*n, n) - binomial(2*n, n+k+1). - _Peter Luschny_, Dec 21 2017

%e The triangle starts:

%e                                 1

%e                               1, 2

%e                             2, 5, 6

%e                          5, 14, 19, 20

%e                        14, 42, 62, 69, 70

%e                   42, 132, 207, 242, 251, 252

%e                132, 429, 704, 858, 912, 923, 924

%p A050157 := (n, k) -> binomial(2*n, n) - binomial(2*n, n+k+1):

%p seq(seq(A050157(n,k), k=0..n), n=0..10); # _Peter Luschny_, Dec 21 2017

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

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

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

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

%Y Row sums are A296771.

%Y Cf. A039599, A050163.

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_