A096466 Triangle T(n,k), read by rows, formed by setting all entries in the zeroth column ((n,0) entries) and the main diagonal ((n,n) entries) to powers of 2 with all other entries formed by the recursion T(n,k) = T(n-1,k) + T(n,k-1).

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, 128, 254, 490, 892, 1476, 2092, 2156, 128, 256, 510, 1000, 1892, 3368, 5460, 7616, 7744, 256, 512, 1022, 2022, 3914, 7282, 12742, 20358, 28102, 28358, 512

Offset: 0

Author

Gerald McGarvey, Aug 12 2004

Keywords

• nonn
• tabl

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)

Examples

			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)
		

Crossrefs

Cf. A000918, A082590.

Programs

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

Extensions

Offset changed to 0 by Petros Hadjicostas, Aug 06 2020