A152400 Triangle T, read by rows, where column k of T = column 0 of matrix power T^(k+1) for k>0, with column 0 of T = unsigned column 0 of T^-1 (shifted).

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

Offset

0,4

Links

Table of n, a(n) for n=0..54.

Formula

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.

Example

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.

Prog

(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))); ))

Crossrefs

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

Cf. related triangles: A152405, A127714.

Sequence in context: A161133 A112911 A152405 * A291978 A342217 A111548

Adjacent sequences: A152397 A152398 A152399 * A152401 A152402 A152403

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 05 2008

Status

approved