OFFSET
0,8
COMMENTS
Rows are transforms of k^n, k>=0, under the matrix A107131. As a number triangle, with T(n,k)=if(k<=n,sum{j=0..n-k, (1/(j+1))C(j+1,n-k-j+1)C(n-k,j)k^j},0), row sums are A107268 and diagonal sums are A107269. Rows are series reversions of x/(1+kx+kx^2), k>=0. Conjecture: rows count weighted Motzkin paths.
Row k counts colored Motzkin paths, where H(1,0) and U(1,1) each have k colors and D(1,-1) one color. - Paul Barry, May 16 2005
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
FORMULA
Number array T(n,k) = Sum_{j=0..k} n^j * binomial(k,j) * binomial(j+1,k-j+1)/(j+1).
G.f. of row k: 1/(1 - k*x - k*x^2/(1 - k*x - k*x^2/(1 - k*x - k*x^2/(1 - k*x - k*x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017
From Seiichi Manyama, May 05 2019: (Start)
T(n,k) = Sum_{j=0..floor(k/2)} n^(k-j) * binomial(k,2*j) * binomial(2*j,j)/(j+1) = Sum_{j=0..floor(k/2)} n^(k-j) * binomial(k,2*j) * A000108(j).
(k+2) * T(n,k) = n * (2*k+1) * T(n,k-1) - n * (n-4) * (k-1) * T(n,k-2). (End)
EXAMPLE
Array begins
1, 0, 0, 0, 0, 0, 0, ...
1, 1, 2, 4, 9, 21, 51, ...
1, 2, 6, 20, 72, 272, 1064, ...
1, 3, 12, 54, 261, 1323, 6939, ...
1, 4, 20, 112, 672, 4224, 27456, ...
1, 5, 30, 200, 1425, 10625, 81875, ...
1, 6, 42, 324, 2664, 22896, 203256, ...
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, May 15 2005
STATUS
approved