OFFSET
0,2
COMMENTS
T(n,k) = T(n-1,k) + T(n,k-1) for n >= 2 and 1 <= k <= n - 1 with T(n,0) = T(n,n) = 2^n for n >= 0.
The n-th row sum equals A082590(n), which is the expansion of 1/(1 - 2*x)/sqrt(1 - 4*x) and equals 2^n * JacobiP(n, 1/2, -1-n, 3).
First column is T(n,1) = A000918(n+1) = 2^(n+1) - 2.
From Petros Hadjicostas, Aug 06 2020: (Start)
T(n,2) = 2^(n+2) - 2*n - 8 for n >= 2.
T(n+1,n) = 2^n + Sum_{k=0..n} T(n,k) = 2^n + A082590(n).
Bivariate o.g.f.: ((1 - x)*(1 - y)/(1 - 2*x) - x*y/sqrt(1 - 4*x*y))/((1 - 2*x*y)*(1 - x - y)). (End)
EXAMPLE
From Petros Hadjicostas, Aug 06 2020: (Start)
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
1;
2, 2;
4, 6, 4;
8, 14, 18, 8;
16, 30, 48, 56, 16;
32, 62, 110, 166, 182, 32;
64, 126, 236, 402, 584, 616, 64;
... (End)
PROG
(PARI) T(n, k) = if ((k==0) || (n==k), 2^n, if ((n<0) || (k<0), 0, if (n>k, T(n-1, k) + T(n, k-1), 0)));
for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Aug 07 2020
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gerald McGarvey, Aug 12 2004
EXTENSIONS
Offset changed to 0 by Petros Hadjicostas, Aug 06 2020
STATUS
approved