

A152391


Triangle T, read by rows, where the matrix square T^2 results in shifting T right one column to drop the secondary diagonal.


8



1, 1, 1, 4, 2, 1, 18, 6, 3, 1, 96, 28, 8, 4, 1, 580, 150, 40, 10, 5, 1, 3852, 930, 216, 54, 12, 6, 1, 27678, 6286, 1386, 294, 70, 14, 7, 1, 212224, 46120, 9552, 1960, 384, 88, 16, 8, 1, 1722312, 359946, 71820, 13770, 2664, 486, 108, 18, 9, 1, 14685140, 2973650, 571440, 106290, 19060, 3510, 600, 130, 20, 10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,4


LINKS



FORMULA

T(n, k) = Sum_{j=k+1..n} T(n,j) * T(j,k+1) for n > k+1 >= 1 with T(n+1,n)=n+1 and T(n,n)=1 for n >= 0.


EXAMPLE

Triangle T begins:
1;
1, 1;
4, 2, 1;
18, 6, 3, 1;
96, 28, 8, 4, 1;
580, 150, 40, 10, 5, 1;
3852, 930, 216, 54, 12, 6, 1;
27678, 6286, 1386, 294, 70, 14, 7, 1;
212224, 46120, 9552, 1960, 384, 88, 16, 8, 1;
1722312, 359946, 71820, 13770, 2664, 486, 108, 18, 9, 1;
14685140, 2973650, 571440, 106290, 19060, 3510, 600, 130, 20, 10, 1; ...
Illustrate recurrence by products of row and column vectors:
T(4,1) = [8,4,1]*[1,3,8]~ = 8*1 + 4*3 + 1*8 = 28;
T(6,0) = [930,216,54,12,6,1]*[1,2,6,28,150,930]~ = 3852;
T(7,0) = [6286,1386,294,70,14,7,1]*[1,2,6,28,150,930,6286]~ = 27678.
T(8,1) = [9552,1960,384,88,16,8,1]*[1,3,8,40,216,1386,9552]~ = 46120.
T(9,3) = [2664,486,108,18,9,1]*[1,5,12,70,384,2664]~ = 13770.
Matrix square T^2 begins:
1;
2, 1;
10, 4, 1;
54, 18, 6, 1;
324, 96, 28, 8, 1;
2130, 580, 150, 40, 10, 1;
15102, 3852, 930, 216, 54, 12, 1;
114282, 27678, 6286, 1386, 294, 70, 14, 1; ...
which equals T shifted right one column with the secondary diagonal dropped.


PROG

(PARI) {T(n, k) = if(n==k, 1, if(n==k+1, n, sum(j=k+1, n, T(n, j)*T(j, k+1) )))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) /* Build an N X N Matrix (informal) */
{M = matrix(N, N, n, k, if(n==k, 1, if(n==k+1, n)) ); }
{T(n, k) = M[n+1, k+1] = if(n==k, 1, if(n==k+1, n, sum(j=k+1, n, T(n, j) * M[j+1, k+2] )))}
for(n=0, N, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Jan 13 2016


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



