A135902 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^3 shift right.
1, 1, 1, 3, 2, 1, 15, 8, 3, 1, 102, 47, 15, 4, 1, 860, 356, 102, 24, 5, 1, 8548, 3252, 860, 186, 35, 6, 1, 97094, 34448, 8548, 1736, 305, 48, 7, 1, 1234324, 412546, 97094, 18754, 3130, 465, 63, 8, 1, 17302880, 5488222, 1234324, 228658, 36630, 5212, 672, 80, 9, 1
Offset: 0
Examples
Triangle T begins: 1; 1, 1; 3, 2, 1; 15, 8, 3, 1; 102, 47, 15, 4, 1; 860, 356, 102, 24, 5, 1; 8548, 3252, 860, 186, 35, 6, 1; 97094, 34448, 8548, 1736, 305, 48, 7, 1; 1234324, 412546, 97094, 18754, 3130, 465, 63, 8, 1; 17302880, 5488222, 1234324, 228658, 36630, 5212, 672, 80, 9, 1; ... where column k of T = column 0 of T^(k+1) with column 0 of T = column 0 of T^3 shift right. Matrix square of T, T^2, begins: 1; 2, 1; 8, 4, 1; 47, 22, 6, 1; 356, 156, 42, 8, 1; 3252, 1343, 351, 68, 10, 1; 34448, 13493, 3415, 656, 100, 12, 1; ... where column 0 of T^2 = column 1 of T. Matrix cube of T, T^3, begins: 1; 3, 1; 15, 6, 1; 102, 42, 9, 1; 860, 351, 81, 12, 1; 8548, 3415, 807, 132, 15, 1; 97094, 37795, 8967, 1530, 195, 18, 1; ... where column 0 of T^3 = column 2 of T = column 0 of T shift left; also, column 1 of T^3 = column 2 of T^2.
Programs
-
PARI
T(n, k)=if(k>n || n<0 || k<0, 0, if(k==n,1,if(k==0, T(n+1,2), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ); )
Formula
Column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.
Comments