A193593 Augmentation of the triangle A193592. See Comments.

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 31, 40, 23, 1, 15, 75, 166, 187, 105, 1, 21, 155, 530, 958, 993, 549, 1, 28, 287, 1415, 3786, 5988, 5865, 3207, 1, 36, 490, 3311, 12441, 28056, 40380, 37947, 20577, 1, 45, 786, 7000, 35469, 109451, 217720, 292092

Offset: 0

Author

Clark Kimberling, Jul 31 2011

Keywords

• nonn
• tabl

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Regarding A193592, (column 1)=A014616, (column 2)=A090809, (right edge)=A113227.

Examples

			First 5 rows:
1
1...1
1...3...2
1...6...10...6
1...10..31...40...23
Rows reversed as in Callan's n-edge increasing ordered trees with outdegree k:
 1
0      1
0      1      1
0      2      3      1
0      6     10      6      1
0     23     40     31     10      1
0    105    187    166     75     15      1
0    549    993    958    530    155     21     1
0   3207   5865   5988   3786   1415    287    28    1
0  20577  37947  40380  28056  12441   3311   490   36   1
0 143239 265901 292092 217720 109451  35469  7000  786  45 1
		

Links

• D. Callan, A bijection to count (1-23-4)-avoiding permutations, arXiv:1008.2375 (rows reversed)

Crossrefs

Cf. A193091, A193592, A113227 (row sums and diagonal), A090809 (3rd col).

Programs

Mathematica

p[n_, 0] := 1; p[n_, k_] := n + 1 - k /; k > 0;
Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A193592 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 12}]]  (* A193593 *)
Flatten[Table[v[n], {n, 0, 10}]]