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 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). 17
 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, 57, 36, 35, 12, 7, 1, 44, 86, 84, 96, 50, 48, 14, 8, 1, 73, 130, 192, 136, 145, 66, 63, 16, 9, 1, 116, 250, 270, 356, 200, 204, 84, 80, 18, 10, 1, 191, 364, 567, 476, 590, 276, 273 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. LINKS FORMULA 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. EXAMPLE Triangle T begins: 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, 57, 36, 35, 12, 7, 1; 44, 86, 84, 96, 50, 48, 14, 8, 1; 73, 130, 192, 136, 145, 66, 63, 16, 9, 1; 116, 250, 270, 356, 200, 204, 84, 80, 18, 10, 1; 191, 364, 567, 476, 590, 276, 273, 104, 99, 20, 11, 1; 294, 696, 780, 1060, 760, 906, 364, 352, 126, 120, 22, 12, 1; ... The matrix square of T, T^2, equals the even-indexed diagonal bisection of T, or T^2 = A118040 = 1; 2, 1; 6, 4, 1; 16, 14, 6, 1; 44, 44, 24, 8, 1; 116, 130, 84, 36, 10, 1; 294, 364, 270, 136, 50, 12, 1; 748, 990, 780, 476, 200, 66, 14, 1; ... Let U = SHIFT_UP(T), which shifts each column of T up 1 row and drops the main diagonal, so that U = 1; 2, 2; 3, 4, 3; 6, 8, 6, 4; 9, 14, 15, 8, 5; 16, 28, 24, 24, 10, 6; ... Then the matrix product T*U equals the odd-indexed diagonal bisection of T, or T*U = A118045 = 1; 3, 2; 9, 8, 3; 26, 28, 15, 4; 73, 86, 57, 24, 5; 191, 250, 192, 96, 35, 6; 500, 696, 567, 356, 145, 48, 7; 1234, 1824, 1683, 1060, 590, 204, 63, 8; ... Thus interleaving diagonals of T^2 and T*U forms T. MAPLE {T(n, k)=if(n

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Last modified January 21 05:39 EST 2020. Contains 331104 sequences. (Running on oeis4.)