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%I #10 Jun 26 2015 22:37:00
%S 1,1,1,4,2,1,28,10,3,1,268,78,18,4,1,3164,798,156,28,5,1,43672,9874,
%T 1714,268,40,6,1,682632,141282,22368,3164,420,54,7,1,11834536,2273730,
%U 333910,43672,5320,618,70,8,1,224283416,40400466,5566728,682632,77720
%N Triangle T, read by rows, where column k of T = column 0 of T^(k+1) for k>0, with column 0 of T = column 0 of T^4 shift right.
%F Column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.
%e Triangle T begins:
%e 1;
%e 1, 1;
%e 4, 2, 1;
%e 28, 10, 3, 1;
%e 268, 78, 18, 4, 1;
%e 3164, 798, 156, 28, 5, 1;
%e 43672, 9874, 1714, 268, 40, 6, 1;
%e 682632, 141282, 22368, 3164, 420, 54, 7, 1;
%e 11834536, 2273730, 333910, 43672, 5320, 618, 70, 8, 1;
%e 224283416, 40400466, 5566728, 682632, 77720, 8378, 868, 88, 9, 1; ...
%e where column k of T = column 0 of T^(k+1)
%e with column 0 of T = column 0 of T^4 shift right:
%e column 1 of T = column 0 of T^2;
%e column 2 of T = column 0 of T^3;
%e column 3 of T = column 0 of T^4.
%e Matrix square of T, T^2, begins:
%e 1;
%e 2, 1;
%e 10, 4, 1;
%e 78, 26, 6, 1;
%e 798, 232, 48, 8, 1;
%e 9874, 2578, 486, 76, 10, 1;
%e 141282, 33764, 5888, 864, 110, 12, 1;
%e 2273730, 503910, 82210, 11396, 1390, 150, 14, 1; ...
%e where column k of T^2 = column 1 of T^(k+1):
%e column 0 of T^2 = column 1 of T;
%e column 2 of T^2 = column 1 of T^3;
%e column 3 of T^2 = column 1 of T^4.
%e Matrix cube of T, T^3, begins:
%e 1;
%e 3, 1;
%e 18, 6, 1;
%e 156, 48, 9, 1;
%e 1714, 486, 90, 12, 1;
%e 22368, 5888, 1050, 144, 15, 1;
%e 333910, 82210, 14046, 1908, 210, 18, 1;
%e 5566728, 1289928, 211182, 28072, 3120, 288, 21, 1; ...
%e where column k of T^3 = column 2 of T^(k+1):
%e column 0 of T^3 = column 2 of T;
%e column 1 of T^3 = column 2 of T^2;
%e column 3 of T^3 = column 2 of T^4.
%e Matrix 4th power of T, T^4, begins:
%e 1;
%e 4, 1;
%e 28, 8, 1;
%e 268, 76, 12, 1;
%e 3164, 864, 144, 16, 1;
%e 43672, 11396, 1908, 232, 20, 1;
%e 682632, 170000, 28072, 3520, 340, 24, 1;
%e 11834536, 2814832, 454848, 57408, 5820, 468, 28, 1; ...
%e where column k of T^4 = column 3 of T^(k+1):
%e column 0 of T^4 = column 3 of T = column 0 of T shift left;
%e column 1 of T^4 = column 3 of T^2;
%e column 2 of T^4 = column 3 of T^3.
%o (PARI) {T(n, k) = if(k>n||k<0, 0, if(k==n, 1, if(k==0, T(n+2, 3), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ); )}
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o (PARI) /* Column k of T = column 0 of T^(k+1): */
%o {T(n, k) = local(M=if(n==0,Mat(1),matrix(n,n,r,c,if(r>=c,T(r-1,c-1))))); if(k==n, 1, if(k==0, (M^4)[n, 1],(M^(k+1))[n-k+1, 1]))}
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%Y Cf. columns: A138272, A138273, A138274; central terms: A138275; variants: A091351, A094587, A135902.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Mar 11 2008
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