A152391 Triangle T, read by rows, where the matrix square T^2 results in shifting T right one column to drop the secondary diagonal.

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

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..350

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

Cf. columns: A152392, A152393, A152394; A152395.

Cf. A109152.

Sequence in context: A269736 A264535 A256039 * A144088 A039948 A111536

Adjacent sequences: A152388 A152389 A152390 * A152392 A152393 A152394

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 11 2008

Status

approved