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%I #4 Jun 14 2017 00:32:23
%S 1,1,1,2,2,1,12,4,3,1,132,30,6,4,1,2200,384,54,8,5,1,50020,7300,780,
%T 84,10,6,1,1458576,186620,16500,1344,120,12,7,1,52443832,6046600,
%U 465240,31120,2100,162,14,8,1,2262992496,239321824,16486512,955000,52600,3072
%N Triangle, read by rows, where diagonal m of T equals diagonal m-1 of matrix power T^m for m>1: T(n,k) = [T^(n-k)](n-1,k) for n>=k>0, with T(n,n)=1 and T(n+1,n)=n+1 for n>=0.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 12, 4, 3, 1;
%e 132, 30, 6, 4, 1;
%e 2200, 384, 54, 8, 5, 1;
%e 50020, 7300, 780, 84, 10, 6, 1;
%e 1458576, 186620, 16500, 1344, 120, 12, 7, 1;
%e 52443832, 6046600, 465240, 31120, 2100, 162, 14, 8, 1;
%e 2262992496, 239321824, 16486512, 955000, 52600, 3072, 210, 16, 9, 1; ...
%e Matrix square, T^2, begins:
%e 1;
%e 2, 1;
%e 6, 4, 1;
%e 34, 14, 6, 1;
%e 354, 88, 24, 8, 1;
%e 5648, 1058, 162, 36, 10, 1; ...
%e where diagonal 1 of T^2 = diagonal 2 of T: [2,4,6,8,10,...].
%e Matrix cube, T^3, begins:
%e 1;
%e 3, 1;
%e 12, 6, 1;
%e 72, 30, 9, 1;
%e 718, 198, 54, 12, 1;
%e 10982, 2210, 384, 84, 15, 1; ...
%e where diagonal 2 of T^3 = diagonal 3 of T: [12,30,54,84,...].
%e Matrix fourth power, T^4, begins:
%e 1;
%e 4, 1;
%e 20, 8, 1;
%e 132, 52, 12, 1;
%e 1300, 384, 96, 16, 1;
%e 19148, 4148, 780, 152, 20, 1;
%e 394412, 67664, 9072, 1344, 220, 24, 1; ...
%e where diagonal 3 of T^4 = diagonal 4 of T: [132,384,780,1344,..].
%o (PARI) T(n,k)=local(M=matrix(n,n,r,c,if(r>=c,T(r-1,c-1)))); if(n<k || k<0,0,if(n==k,1,if(n==k+1,n,(M^(n-k))[n,k+1])))
%Y Cf. A132472 (column 0), A132473 (column 1).
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Aug 22 2007
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