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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.

%I A131909 #7 Jun 14 2017 00:31:45
%S A131909 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,
%T A131909 13,16,18,21,25,25,14,22,21,26,29,34,39,46,46,25,39,36,43,47,55,63,73,
%U A131909 85,85,46,71,64,75,79,90,102,118,136,158,158,85,131,117,135,139,154,169
%N 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.
%C A131909 A119262(n) is the number of B-trees of order infinity with n leaves.
%F A131909 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)).
%e A131909 Triangle begins:
%e A131909 1;
%e A131909 1, 1;
%e A131909 1, 1, 2;
%e A131909 2, 1, 2, 3;
%e A131909 3, 2, 3, 3, 5;
%e A131909 5, 3, 5, 5, 6, 8;
%e A131909 8, 5, 8, 8, 10, 11, 14;
%e A131909 14, 8, 13, 13, 16, 18, 21, 25;
%e A131909 25, 14, 22, 21, 26, 29, 34, 39, 46;
%e A131909 46, 25, 39, 36, 43, 47, 55, 63, 73, 85;
%e A131909 85, 46, 71, 64, 75, 79, 90, 102, 118, 136, 158;
%e A131909 158, 85, 131, 117, 135, 139, 154, 169, 192, 220, 254, 294; ...
%e A131909 Illustrate T(n,k) = T(n-1,k-2) + T(n-1,k-1):
%e A131909 T(5,3) = T(4,1) + T(4,2) = 2 + 3 = 5;
%e A131909 T(6,4) = T(5,2) + T(5,3) = 5 + 5 = 10;
%e A131909 T(8,3) = T(7,1) + T(7,2) = 8 +13 = 21.
%o A131909 (PARI) T(n,k)=if(k<0 || n<k,0,if(n==0 && k==0,1,if(k==0,T(n-1,n-1),T(n-1,k-2)+T(n-1,k-1))))
%Y A131909 Cf. A119262 (columns 0, 1 and main diagonal); A131910 (central terms).
%K A131909 nonn,tabl
%O A131909 0,6
%A A131909 _Paul D. Hanna_, Jul 26 2007