A130580 - OEIS

%I #4 Jun 14 2017 00:31:03

%S 1,1,1,1,1,1,1,2,1,1,1,3,3,1,1,1,5,6,4,1,1,1,9,13,10,5,1,1,1,18,30,26,

%T 15,6,1,1,1,40,76,71,45,21,7,1,1,1,98,211,210,140,71,28,8,1,1,1,263,

%U 638,673,466,246,105,36,9,1,1,1,769,2088,2328,1665,902,399,148,45,10,1,1,1

%N Triangle, read by rows, where column k = column 0 (shifted) of matrix power T^(k+1) for k>=0, with T(0,0)=1.

%e Triangle T begins:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 2, 1, 1;

%e 1, 3, 3, 1, 1;

%e 1, 5, 6, 4, 1, 1;

%e 1, 9, 13, 10, 5, 1, 1;

%e 1, 18, 30, 26, 15, 6, 1, 1;

%e 1, 40, 76, 71, 45, 21, 7, 1, 1;

%e 1, 98, 211, 210, 140, 71, 28, 8, 1, 1;

%e 1, 263, 638, 673, 466, 246, 105, 36, 9, 1, 1;

%e 1, 769, 2088, 2328, 1665, 902, 399, 148, 45, 10, 1, 1; ...

%e Matrix square T^2 begins:

%e 1;

%e 2, 1;

%e 3, 2, 1;

%e 5, 5, 2, 1;

%e 9, 11, 7, 2, 1;

%e 18, 27, 19, 9, 2, 1;

%e 40, 71, 57, 29, 11, 2, 1;

%e 98, 202, 180, 101, 41, 13, 2, 1; ...

%e where column 0 of T^2 equals column 1 of T (shifted).

%e Matrix cube T^3 begins:

%e 1;

%e 3, 1;

%e 6, 3, 1;

%e 13, 9, 3, 1;

%e 30, 25, 12, 3, 1;

%e 76, 75, 40, 15, 3, 1;

%e 211, 238, 144, 58, 18, 3, 1; ...

%e where column 0 of T^3 equals column 2 of T (shifted).

%e Matrix power T^4 begins:

%e 1;

%e 4, 1;

%e 10, 4, 1;

%e 26, 14, 4, 1;

%e 71, 46, 18, 4, 1;

%e 210, 159, 70, 22, 4, 1; ...

%e where column 0 of T^4 equals column 3 of T (shifted).

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

%Y Columns: A130581, A130582, A130583.

%K nonn,tabl

%O 0,8

%A _Paul D. Hanna_, Jun 05 2007