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A193593 Augmentation of the triangle A193592. See Comments.

%I A193593 #8 Dec 05 2022 09:43:08
%S A193593 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,
%T A193593 958,993,549,1,28,287,1415,3786,5988,5865,3207,1,36,490,3311,12441,
%U A193593 28056,40380,37947,20577,1,45,786,7000,35469,109451,217720,292092
%N A193593 Augmentation of the triangle A193592.  See Comments.
%C A193593 For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%C A193593 Regarding A193592, (column 1)=A014616, (column 2)=A090809, (right edge)=A113227.
%H A193593 D. Callan, <a href="https://arxiv.org/abs/1008.2375">A bijection to count (1-23-4)-avoiding permutations</a>, arXiv:1008.2375 (rows reversed)
%e A193593 First 5 rows:
%e A193593 1
%e A193593 1...1
%e A193593 1...3...2
%e A193593 1...6...10...6
%e A193593 1...10..31...40...23
%e A193593 Rows reversed as in Callan's n-edge increasing ordered trees with outdegree k:
%e A193593  1
%e A193593 0      1
%e A193593 0      1      1
%e A193593 0      2      3      1
%e A193593 0      6     10      6      1
%e A193593 0     23     40     31     10      1
%e A193593 0    105    187    166     75     15      1
%e A193593 0    549    993    958    530    155     21     1
%e A193593 0   3207   5865   5988   3786   1415    287    28    1
%e A193593 0  20577  37947  40380  28056  12441   3311   490   36   1
%e A193593 0 143239 265901 292092 217720 109451  35469  7000  786  45 1
%t A193593 p[n_, 0] := 1; p[n_, k_] := n + 1 - k /; k > 0;
%t A193593 Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A193592 *)
%t A193593 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193593 TableForm[m[4]]
%t A193593 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193593 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193593 v[n_] := v[n - 1].m[n]
%t A193593 TableForm[Table[v[n], {n, 0, 12}]]  (* A193593 *)
%t A193593 Flatten[Table[v[n], {n, 0, 10}]]
%Y A193593 Cf. A193091, A193592, A113227 (row sums and diagonal), A090809 (3rd col).
%K A193593 nonn,tabl
%O A193593 0,5
%A A193593 _Clark Kimberling_, Jul 31 2011