OFFSET
0,1
FORMULA
From Jianing Song, May 30 2022: (Start)
T(n,k) = 2 if k = n, otherwise A052509(2n,n+1+k) + A052509(2n,n-k) = 2^(n-1-k) + Sum_{m=0..n-k} binomial(n+k,m) = 2^(n-1-k) + 2^(n+k) - Sum_{m=0..2*k-1} binomial(n+k,m).
T(n,k) = [x^n*y^(n-k)] (1-x*y) * ((1+y-x*y^2)/((1-x*y^2)*((1-x*y)^2-x)) + (1+y-x*y)/((1-x)*((1-x*y)^2-x*y^2))). (End)
EXAMPLE
Rows:
2;
3, 2;
6, 5, 2;
12, 13, 7, 2;
...
PROG
(PARI) T(n, k) = if(k==n, 2, 2^(n-1-k) + sum(m=0, n-k, binomial(n+k, m))) \\ Jianing Song, May 30 2022
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved