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