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%I #7 Jun 14 2017 01:08:10
%S 1,1,1,2,2,1,6,6,4,1,26,26,18,8,1,162,162,114,54,16,1,1454,1454,1030,
%T 506,162,32,1,18854,18854,13394,6666,2274,486,64,1,354258,354258,
%U 251962,126134,43798,10346,1458,128,1,9671546,9671546,6882102,3453110,1210226
%N Triangle T, read by rows, where the n-th diagonal of T equals the BINOMIAL transform of the (n-1)-th diagonal of T^2 for n>=1, with the zeroth diagonal set to all 1's and where T^2 denotes the matrix square of T.
%F T(n,k) = Sum_{j=0..k} C(k,j)*[T^2](n-k+j-1,j) for n>k>=0, with T(n,n)=1, for n>=0.
%e Triangle T begins:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 6, 6, 4, 1;
%e 26, 26, 18, 8, 1;
%e 162, 162, 114, 54, 16, 1;
%e 1454, 1454, 1030, 506, 162, 32, 1;
%e 18854, 18854, 13394, 6666, 2274, 486, 64, 1;
%e 354258, 354258, 251962, 126134, 43798, 10346, 1458, 128, 1; ...
%e Matrix square T^2 begins:
%e 1;
%e 2, 1;
%e 6, 4, 1;
%e 26, 20, 8, 1;
%e 162, 136, 68, 16, 1;
%e 1454, 1292, 732, 236, 32, 1;
%e 18854, 17400, 10648, 4036, 836, 64, 1; ...
%e where the BINOMIAL transform of diagonal 2 of T^2:
%e BINOMIAL[6,20,68,236,836,3020,11108,41516,...]
%e equals: [6,26,114,506,2274,10346,47634,221786,...]
%e which is diagonal 3 of T.
%e Specific examples:
%e T(4,1) = [T^2](2,0) + [T^2](3,1) = 6 + 20 = 26;
%e T(4,2) = [T^2](1,0) + 2*[T^2](2,1) + [T^2](3,2) = 2 + 2*4 + 8 = 18;
%e T(5,2) = [T^2](2,0) + 2*[T^2](3,1) + [T^2](4,2) = 6 + 2*20 + 68 = 114;
%e T(5,3) = [T^2](1,0) + 3*[T^2](2,1) + 3*[T^2](3,2) + [T^2](4,3) = 2 + 3*4 + 3*8 + 16 = 54.
%o (PARI) T(n,k)=if(n<k || k<0,0,if(n==k,1,if(n==k+1,2^k,if(k==1,T(n,0), sum(j=0,k,binomial(k,j)*sum(i=0,n-k+j-1,T(n-k+j-1,i)*T(i,j)))))))
%Y Cf. A141713 (column 0), A141714 (column 2); A141715 (T^2), A141716.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Jul 01 2008
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