%I #17 Jul 03 2024 20:01:51

%S 1,1,1,3,2,1,11,5,3,1,50,20,7,4,1,257,94,31,9,5,1,1467,507,150,44,11,

%T 6,1,9081,3009,853,218,59,13,7,1,60272,19350,5251,1307,298,76,15,8,1,

%U 424514,132920,35109,8313,1881,390,95,17,9,1,3151226,966962,249332

%N Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n that are to the right of T(n,k) with the vector of terms in column k that are above T(n,k): T(n,k) = Sum_{j=0..n-k-1} T(n,j+k+1)*T(j+k,k) for n > k+1 > 0, with T(n,n) = 1 and T(n,n-1) = n (n>=1).

%C Triangle A115085 is the dual of this triangle.

%H Paul D. Hanna, <a href="/A115080/b115080.txt">Table of n, a(n) for n = 0..350, as a flattened triangle of rows 0..25.</a>

%e T(n,k) = [T(n,k+1),T(n,k+2),...,T(n,n)]*[T(k,k),T(k+1,k),...,T(n-1,k)]:

%e T(3,0) = [5,3,1]*[1,1,3] = 5*1 + 3*1 + 1*3 = 11;

%e T(4,1) = [7,4,1]*[1,2,5] = 7*1 + 4*2 + 1*5 = 20;

%e T(5,1) = [31,9,5,1]*[1,2,5,20] = 31*1 + 9*2 + 5*5 + 1*20 = 94;

%e T(6,2) = [44,11,6,1]*[1,3,7,31] = 44*1 + 11*3 + 6*7 + 1*31 = 150.

%e Triangle begins:

%e 1;

%e 1, 1;

%e 3, 2, 1;

%e 11, 5, 3, 1;

%e 50, 20, 7, 4, 1;

%e 257, 94, 31, 9, 5, 1;

%e 1467, 507, 150, 44, 11, 6, 1;

%e 9081, 3009, 853, 218, 59, 13, 7, 1;

%e 60272, 19350, 5251, 1307, 298, 76, 15, 8, 1;

%e 424514, 132920, 35109, 8313, 1881, 390, 95, 17, 9, 1;

%e 3151226, 966962, 249332, 57738, 12315, 2587, 494, 116, 19, 10, 1;

%e 24510411, 7396366, 1873214, 422948, 88737, 17377, 3437, 610, 139, 21, 11, 1;

%e ...

%o (PARI) {T(n,k)=if(n==k,1,if(n==k+1,n, sum(j=0,n-k-1,T(n,j+k+1)*T(j+k,k))))}

%o for(n=0,12,for(k=0,n, print1(T(n,k),", "));print(""))

%Y Cf. A115081 (column 0), A115082 (column 1), A115083 (column 2), A115084 (row sums); A115085 (dual triangle).

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Jan 13 2006