Search: id:A118032
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%I A118032
%S A118032 1,1,1,2,2,1,3,4,3,1,6,8,6,4,1,9,14,15,8,5,1,16,28,24,24,10,6,1,26,44,
%T A118032 57,36,35,12,7,1,44,86,84,96,50,48,14,8,1,73,130,192,136,145,66,63,16,
               9,
%U A118032 1,116,250,270,356,200,204,84,80,18,10,1,191,364,567,476,590,276,273
%N A118032 Triangle T, read by rows, such that diagonal 2n of T equals diagonal 
               n of T^2 and diagonal 2n+1 of T equals diagonal n of T*U: [T^2](n,
               k) = T(2n-k,k) and [T*U](n,k) = T(2n+1-k,k) for n>=k, k>=0, where 
               U = SHIFT_UP(T).
%C A118032 The diagonal bisections of this triangle T forms the diagonals of T^2 
               and T*U, where U = SHIFT_UP(T) indicates that U results from shifting 
               each column of T up 1 row, dropping the main diagonal of all 1's.
%F A118032 T(2n-k,k) = Sum_{j=k..n} T(n,j)*T(j,k) = [T^2](n,k) for n>=k; odd-indexed 
               diagonals: T(2n+1-k,k) = Sum_{j=k..n} T(n,j)*T(j+1,k) = [T*U](n,k) 
               for n>=k; with T(n+1,n)=n+1, T(n,n)=1.
%e A118032 Triangle T begins:
%e A118032 1;
%e A118032 1, 1;
%e A118032 2, 2, 1;
%e A118032 3, 4, 3, 1;
%e A118032 6, 8, 6, 4, 1;
%e A118032 9, 14, 15, 8, 5, 1;
%e A118032 16, 28, 24, 24, 10, 6, 1;
%e A118032 26, 44, 57, 36, 35, 12, 7, 1;
%e A118032 44, 86, 84, 96, 50, 48, 14, 8, 1;
%e A118032 73, 130, 192, 136, 145, 66, 63, 16, 9, 1;
%e A118032 116, 250, 270, 356, 200, 204, 84, 80, 18, 10, 1;
%e A118032 191, 364, 567, 476, 590, 276, 273, 104, 99, 20, 11, 1;
%e A118032 294, 696, 780, 1060, 760, 906, 364, 352, 126, 120, 22, 12, 1; ...
%e A118032 The matrix square of T, T^2, equals the even-indexed
%e A118032 diagonal bisection of T, or T^2 = A118040 =
%e A118032 1;
%e A118032 2, 1;
%e A118032 6, 4, 1;
%e A118032 16, 14, 6, 1;
%e A118032 44, 44, 24, 8, 1;
%e A118032 116, 130, 84, 36, 10, 1;
%e A118032 294, 364, 270, 136, 50, 12, 1;
%e A118032 748, 990, 780, 476, 200, 66, 14, 1; ...
%e A118032 Let U = SHIFT_UP(T), which shifts each column of T up 1 row
%e A118032 and drops the main diagonal, so that U =
%e A118032 1;
%e A118032 2, 2;
%e A118032 3, 4, 3;
%e A118032 6, 8, 6, 4;
%e A118032 9, 14, 15, 8, 5;
%e A118032 16, 28, 24, 24, 10, 6; ...
%e A118032 Then the matrix product T*U equals the odd-indexed
%e A118032 diagonal bisection of T, or T*U = A118045 =
%e A118032 1;
%e A118032 3, 2;
%e A118032 9, 8, 3;
%e A118032 26, 28, 15, 4;
%e A118032 73, 86, 57, 24, 5;
%e A118032 191, 250, 192, 96, 35, 6;
%e A118032 500, 696, 567, 356, 145, 48, 7;
%e A118032 1234, 1824, 1683, 1060, 590, 204, 63, 8; ...
%e A118032 Thus interleaving diagonals of T^2 and T*U forms T.
%p A118032 {T(n,k)=if(n<k|k<0,0,if(n==k,1,if(n==k+1,n, sum(i=k,(n+k)\2,T((n+k)\2,
               i)*T(i+(n+k)%2,k) ))))}
%Y A118032 Columns: A118033, A118034, A118035; A118036 (row sums), A118037 (antidiagonal 
               sums), A118038 (central terms), A118039; diagonal bisections: A118040 
               (T^2), A118045 (T*U).
%Y A118032 Sequence in context: A061987 A105809 A091594 this_sequence A089692 A066201 
               A123956
%Y A118032 Adjacent sequences: A118029 A118030 A118031 this_sequence A118033 A118034 
               A118035
%K A118032 nonn,tabl
%O A118032 0,4
%A A118032 Paul D. Hanna (pauldhanna(AT)juno.com), Apr 10 2006

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