0,4

Column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.

Column k: T(n,k) = Sum_{j=0..n-k} T(n-k,j)*T(j+k-1,k-1) for n>=k>0.

Column 0: T(n,0) = Sum_{j=1..n} T(n,j)*T(j-1,0) for n>=0.

Triangle T begins:

1;

1, 1;

3, 2, 1;

14, 8, 3, 1;

86, 45, 15, 4, 1;

645, 318, 99, 24, 5, 1;

5662, 2671, 794, 182, 35, 6, 1;

56632, 25805, 7414, 1636, 300, 48, 7, 1;

633545, 280609, 78507, 16844, 2990, 459, 63, 8, 1;

7820115, 3381993, 926026, 194384, 33685, 5026, 665, 80, 9, 1;...

where column k of T = column 0 of T^(k+1) for k>0

and column 0 of T = unsigned column 0 of T^-1 (shifted).

Amazingly, column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.

Matrix inverse T^-1 begins:

1;

-1, 1;

-1, -2, 1;

-3, -2, -3, 1;

-14, -7, -3, -4, 1;

-86, -37, -12, -4, -5, 1;

-645, -252, -71, -18, -5, -6, 1;...

where unsigned column 0 of T^-1 = column 0 of T (shifted).

Matrix square T^2 begins:

1;

2, 1;

8, 4, 1;

45, 22, 6, 1;

318, 152, 42, 8, 1;

2671, 1251, 345, 68, 10, 1;

25805, 11869, 3253, 648, 100, 12, 1;

280609, 126987, 34546, 6898, 1085, 138, 14, 1;...

where column 0 of T^2 = column 1 of T,

and column 2 of T^2 = column 1 of T^3.

Matrix cube T^3 begins:

1;

3, 1;

15, 6, 1;

99, 42, 9, 1;

794, 345, 81, 12, 1;

7414, 3253, 798, 132, 15, 1;

78507, 34546, 8679, 1518, 195, 18, 1;

926026, 407171, 103707, 18734, 2565, 270, 21, 1;...

where column 0 of T^3 = column 2 of T,

and column 3 of T^3 = column 2 of T^4.

Matrix power T^4 begins:

1;

4, 1;

24, 8, 1;

182, 68, 12, 1;

1636, 648, 132, 16, 1;

16844, 6898, 1518, 216, 20, 1;

194384, 81218, 18734, 2912, 320, 24, 1;

2476868, 1047638, 249202, 40932, 4950, 444, 28, 1;...

where column 0 of T^4 = column 3 of T,

and column 2 of T^4 = column 3 of T^3.

Related triangle A127714 begins:

1;

1, 1, 1;

1, 2, 2, 3, 3, 3;

1, 3, 5, 5, 8, 11, 11, 14, 14, 14;

1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86;...

where right border = column 0 of this triangle A152400.

(PARI) {T(n, k)=if(k>n|n<0, 0, if(k==n, 1, if(k==0, sum(j=1, n, T(n, j)*T(j-1, 0)), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ))}

Cf. columns: A127715, A152401, A152402, A152403, A152404.

Cf. related triangles: A152405, A127714.

Sequence in context: A161133 A112911 A152405 this_sequence A111548 A140709 A109282

Adjacent sequences: A152397 A152398 A152399 this_sequence A152401 A152402 A152403

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 05 2008