login
T(n, k) = S(2n+2, n+2, k+2) for 0<=k<=n and n >= 0, array S as in A050157.
2

%I #12 Dec 21 2017 06:19:22

%S 1,3,4,9,14,15,28,48,55,56,90,165,200,209,210,297,572,726,780,791,792,

%T 1001,2002,2639,2912,2989,3002,3003,3432,7072,9620,10880,11320,11424,

%U 11439,11440,11934,25194,35190,40698,42942,43605

%N T(n, k) = S(2n+2, n+2, k+2) for 0<=k<=n and n >= 0, array S as in A050157.

%F T(n, k) = Sum_{0<=j<=k} t(n, j), array t as in A050155.

%F T(n, k) = binomial(2*n+2, n) - binomial(2*n+2, n+k+3). - _Peter Luschny_, Dec 21 2017

%e Triangle starts:

%e 1

%e 3, 4

%e 9, 14, 15

%e 28, 48, 55, 56

%e 90, 165, 200, 209, 210

%e 297, 572, 726, 780, 791, 792

%e 1001, 2002, 2639, 2912, 2989, 3002, 3003

%p A050163 := (n, k) -> binomial(2*n+2, n) - binomial(2*n+2, n+k+3):

%p seq(seq(A050163(n,k), k=0..n), n=0..8); # _Peter Luschny_, Dec 21 2017

%Y T(n, 0) = A000245(n+1).

%Y T(n, 1) = A002057(n).

%Y T(n, n) = A001791(n+1).

%Y Row sums are A000531(n+1).

%Y Cf. A050155, A050157.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_