A131909 Triangle, read by rows, where T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n>=k>1, with T(0,0)=1 and T(n,0) = T(n+1,1) = T(n-1,n-1) for n>0.

1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 3, 3, 5, 5, 3, 5, 5, 6, 8, 8, 5, 8, 8, 10, 11, 14, 14, 8, 13, 13, 16, 18, 21, 25, 25, 14, 22, 21, 26, 29, 34, 39, 46, 46, 25, 39, 36, 43, 47, 55, 63, 73, 85, 85, 46, 71, 64, 75, 79, 90, 102, 118, 136, 158, 158, 85, 131, 117, 135, 139, 154, 169

Offset: 0

Author

Paul D. Hanna, Jul 26 2007

Keywords

• nonn
• tabl

Comments

A119262(n) is the number of B-trees of order infinity with n leaves.

Examples

			Triangle begins:
1;
1, 1;
1, 1, 2;
2, 1, 2, 3;
3, 2, 3, 3, 5;
5, 3, 5, 5, 6, 8;
8, 5, 8, 8, 10, 11, 14;
14, 8, 13, 13, 16, 18, 21, 25;
25, 14, 22, 21, 26, 29, 34, 39, 46;
46, 25, 39, 36, 43, 47, 55, 63, 73, 85;
85, 46, 71, 64, 75, 79, 90, 102, 118, 136, 158;
158, 85, 131, 117, 135, 139, 154, 169, 192, 220, 254, 294; ...
Illustrate T(n,k) = T(n-1,k-2) + T(n-1,k-1):
T(5,3) = T(4,1) + T(4,2) = 2 + 3 = 5;
T(6,4) = T(5,2) + T(5,3) = 5 + 5 = 10;
T(8,3) = T(7,1) + T(7,2) = 8 +13 = 21.
		

Crossrefs

Cf. A119262 (columns 0, 1 and main diagonal); A131910 (central terms).

Programs

PARI

T(n,k)=if(k<0 || n
				

Formula

Row sums equal powers of 2. T(n,0) = A119262(n+1) for n>=0, where g.f. G(x) of A119262 satisfies: G(x) = x + G(x^2/(1-x)).