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 A109152 Triangle T, read by rows, that satisfies: T(n,k) = [T^2](n-1,k) for n>k+1>=1, with T(n,n) = 1 and T(n+1,n) = n+1 for n>=0, where T^2 is the matrix square of T. 7
 1, 1, 1, 2, 2, 1, 6, 4, 3, 1, 22, 14, 6, 4, 1, 94, 56, 24, 8, 5, 1, 450, 262, 102, 36, 10, 6, 1, 2366, 1348, 516, 160, 50, 12, 7, 1, 13450, 7574, 2814, 868, 230, 66, 14, 8, 1, 81802, 45616, 16752, 4968, 1330, 312, 84, 16, 9, 1, 527826, 292442, 106026, 31148, 7930, 1914, 406, 104, 18, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..350 FORMULA T^(m+1) = SHIFT_UP(T^m - T^(m-1)) - D*T^(m-1) for all m where diagonal matrix D = [0, 1, 2, 3, ...] and SHIFT_UP shifts each column up 1 row. Recurrence: T(n,k) = Sum_{j=k..n-1} T(n-1, j) * T(j, k) for k=0..n-2 for n>1 with T(n+1,n) = n+1 and T(n,n) = 1 for n>=0. - Paul D. Hanna, Jan 13 2016 EXAMPLE Triangle T begins: 1; 1, 1; 2, 2, 1; 6, 4, 3, 1; 22, 14, 6, 4, 1; 94, 56, 24, 8, 5, 1; 450, 262, 102, 36, 10, 6, 1; 2366, 1348, 516, 160, 50, 12, 7, 1; 13450, 7574, 2814, 868, 230, 66, 14, 8, 1; 81802, 45616, 16752, 4968, 1330, 312, 84, 16, 9, 1; 527826, 292442, 106026, 31148, 7930, 1914, 406, 104, 18, 10, 1; ... Matrix square T^2 starts:       1;       2       1;       6       4      1;      22      14      6      1;      94      56     24      8     1;     450     262    102     36    10     1;    2366    1348    516    160    50    12    1;   13450    7574   2814    868   230    66   14   1;   81802   45616  16752   4968  1330   312   84  16   1; 527826  292442 106026  31148  7930  1914  406 104  18  1; 3590294 1978788 712260 205888 52130 11820 2632 512 126 20 1; ... which equals SHIFT_UP(T) - D where D is the diagonal matrix [0,1,2,3,...]. PROG (PARI) {T(n, k) = local(M=matrix(n, n, r, c, if(r>=c, T(r-1, c-1)))); if(n

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Last modified August 7 12:19 EDT 2020. Contains 336276 sequences. (Running on oeis4.)