A152391 - OEIS

%I #18 Apr 21 2019 16:21:17

%S 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,

%T 12,6,1,27678,6286,1386,294,70,14,7,1,212224,46120,9552,1960,384,88,

%U 16,8,1,1722312,359946,71820,13770,2664,486,108,18,9,1,14685140,2973650,571440,106290,19060,3510,600,130,20,10,1

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

%H Paul D. Hanna, <a href="/A152391/b152391.txt">Table of n, a(n) for n = 0..350</a>

%F 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.

%e Triangle T begins:

%e 1;

%e 1, 1;

%e 4, 2, 1;

%e 18, 6, 3, 1;

%e 96, 28, 8, 4, 1;

%e 580, 150, 40, 10, 5, 1;

%e 3852, 930, 216, 54, 12, 6, 1;

%e 27678, 6286, 1386, 294, 70, 14, 7, 1;

%e 212224, 46120, 9552, 1960, 384, 88, 16, 8, 1;

%e 1722312, 359946, 71820, 13770, 2664, 486, 108, 18, 9, 1;

%e 14685140, 2973650, 571440, 106290, 19060, 3510, 600, 130, 20, 10, 1; ...

%e Illustrate recurrence by products of row and column vectors:

%e T(4,1) = [8,4,1]*[1,3,8]~ = 8*1 + 4*3 + 1*8 = 28;

%e T(6,0) = [930,216,54,12,6,1]*[1,2,6,28,150,930]~ = 3852;

%e T(7,0) = [6286,1386,294,70,14,7,1]*[1,2,6,28,150,930,6286]~ = 27678.

%e T(8,1) = [9552,1960,384,88,16,8,1]*[1,3,8,40,216,1386,9552]~ = 46120.

%e T(9,3) = [2664,486,108,18,9,1]*[1,5,12,70,384,2664]~ = 13770.

%e Matrix square T^2 begins:

%e 1;

%e 2, 1;

%e 10, 4, 1;

%e 54, 18, 6, 1;

%e 324, 96, 28, 8, 1;

%e 2130, 580, 150, 40, 10, 1;

%e 15102, 3852, 930, 216, 54, 12, 1;

%e 114282, 27678, 6286, 1386, 294, 70, 14, 1; ...

%e which equals T shifted right one column with the secondary diagonal dropped.

%o (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) )))}

%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

%o (PARI) /* Build an N X N Matrix (informal) */

%o {M = matrix(N,N,n,k,if(n==k,1,if(n==k+1,n)) );}

%o {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] )))}

%o for(n=0,N,for(k=0,n,print1(T(n,k),", "));print("")) \\ _Paul D. Hanna_, Jan 13 2016

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

%Y Cf. A109152.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Dec 11 2008