A115085 Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n-1 from T(n-1,k) to T(n-1,n-1) with the vector of terms in column k+1 from T(k+1,k+1) to T(n,k+1): T(n,k) = Sum_{j=0..n-k-1} T(n-1,j+k)*T(j+k+1,k+1) for n>k+1>0, with T(n,n) = 1 and T(n,n-1) = n (n>=1).

1, 1, 1, 3, 2, 1, 12, 5, 3, 1, 58, 21, 7, 4, 1, 321, 102, 32, 9, 5, 1, 1963, 579, 158, 45, 11, 6, 1, 13053, 3601, 933, 226, 60, 13, 7, 1, 92946, 24426, 5939, 1395, 306, 77, 15, 8, 1, 702864, 176858, 41385, 9097, 1977, 398, 96, 17, 9, 1, 5599204, 1359906, 306070

Offset: 0

Author

Paul D. Hanna, Jan 13 2006

Keywords

• nonn
• tabl

Comments

Triangle A115080 is the dual of this triangle.

Examples

			T(n,k)=[T(n-1,k),T(n-1,k+1),..,T(n-1,n-1)]*[T(k+1,k+1),T(k+2,k+1),..,T(n,k+1)]:
12 = [3,2,1]*[1,2,5] = 3*1 + 2*2 + 1*5;
21 = [5,3,1]*[1,3,7] = 5*1 + 3*3 + 1*7;
102 = [21,7,4,1]*[1,3,7,32] = 21*1 + 7*3 + 4*7 + 1*32;
158 = [32,9,5,1]*[1,4,9,45] = 32*1 + 9*4 + 5*9 + 1*45.
Triangle begins:
1;
1, 1;
3, 2, 1;
12, 5, 3, 1;
58, 21, 7, 4, 1;
321, 102, 32, 9, 5, 1;
1963, 579, 158, 45, 11, 6, 1;
13053, 3601, 933, 226, 60, 13, 7, 1;
92946, 24426, 5939, 1395, 306, 77, 15, 8, 1;
702864, 176858, 41385, 9097, 1977, 398, 96, 17, 9, 1;
5599204, 1359906, 306070, 65310, 13195, 2691, 502, 117, 19, 10, 1;
46746501, 10996740, 2403792, 494022, 97701, 18353, 3549, 618, 140, 21, 11, 1;
407019340, 93136545, 19799468, 3970878, 755834, 140178, 24691, 4563, 746, 165, 23, 12, 1; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..405, as a flattened triangle of rows 0..27.

Crossrefs

Cf. A115086 (column 0), A115087 (column 1), A115088 (column 2), A115089 (row sums); A115080 (dual triangle).

Programs

PARI

{T(n,k)=if(n==k,1,if(n==k+1,n, sum(j=0,n-k-1,T(n-1,j+k)*T(j+k+1,k+1))))}
for(n=0,12,for(k=0,n, print1(T(n,k),", "));print(""))