A130528 - OEIS

%I #6 Jun 14 2017 00:31:00

%S 1,1,1,2,2,1,12,8,3,1,156,80,20,4,1,3540,1516,300,40,5,1,123400,46236,

%T 7816,840,70,6,1,6091988,2054980,309268,28816,1960,112,7,1,402900176,

%U 124679524,17129124,1437476,85656,4032,168,8,1,34289884368,9862677332

%N Triangle T, read by rows, where row n+1 of T equals row n of matrix power T^n added to row n of T (shifted right).

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, 2, 1;

%e 12, 8, 3, 1;

%e 156, 80, 20, 4, 1;

%e 3540, 1516, 300, 40, 5, 1;

%e 123400, 46236, 7816, 840, 70, 6, 1;

%e 6091988, 2054980, 309268, 28816, 1960, 112, 7, 1;

%e 402900176, 124679524, 17129124, 1437476, 85656, 4032, 168, 8, 1; ...

%e Matrix square T^2 begins:

%e 1;

%e 2, 1;

%e 6, 4, 1;

%e 38, 22, 6, 1;

%e 480, 232, 52, 8, 1; ...

%e [Row 2 of T] = [row 1 of T^2, 0] + [0, row 1 of T]:

%e [2, 2, 1] = [2, 1, 0] + [0, 1, 1].

%e Matrix cube T^3 begins:

%e 1;

%e 3, 1;

%e 12, 6, 1; ...

%e [Row 3 of T] = [row 2 of T^3, 0] + [0, row 2 of T]:

%e [12, 8, 3, 1] = [12, 6, 1, 0] + [0, 2, 2, 1].

%e Matrix 4th power T^4 begins:

%e 1;

%e 4, 1;

%e 20, 8, 1;

%e 156, 68, 12, 1; ...

%e [Row 4 of T] = [row 3 of T^4, 0] + [0, row 3 of T]:

%e [156, 80, 20, 4, 1] = [156, 68, 12, 1, 0] + [0, 12, 8, 3, 1].

%o (PARI) T(n,k)=local(M=Mat(1));if(n<k || k<0,0,if(n==k,1, M=matrix(n+1,n+1,r,c,if(n==k,1,if(r>=c && r<=n,T(r-1,c-1)))); T(n-1,k-1)+(M^n)[n,k+1]))

%Y Columns: A130529, A130530, A130531.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Jun 02 2007