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.

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

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<k||k<0, 0, if(n==k||n==0, 1, if(n==k+1, n, (M^2)[n, k+1])))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) {T(n, k) = if(n==k, 1, if(n==k+1, n, sum(j=k, n-1, T(n-1, j) * T(j, k) )))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Jan 13 2016

Crossrefs

Cf. A109153 (column 0), A109154 (column 1), A109155 (column 2), A109156 (row sums).

Cf. A152391.

Sequence in context: A178886 A065052 A128308 * A130469 A106381 A064784

Adjacent sequences: A109149 A109150 A109151 * A109153 A109154 A109155

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 20 2005

Status

approved