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%I #55 Oct 14 2022 05:58:13
%S 1,1,0,1,1,0,1,1,1,0,1,1,2,1,0,1,1,2,4,1,0,1,1,2,5,9,1,0,1,1,2,5,13,
%T 21,1,0,1,1,2,5,14,36,51,1,0,1,1,2,5,14,41,104,127,1,0,1,1,2,5,14,42,
%U 125,309,323,1,0,1,1,2,5,14,42,131,393,939,835,1,0
%N Number A(n,k) of ordered rooted trees with n non-root nodes and all outdegrees <= k; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.
%C Also the number of Dyck paths of semilength n with all ascent lengths <= k. A(4,2) = 9: /\/\/\/\, //\\/\/\, /\//\\/\, /\/\//\\, //\/\\/\, //\/\/\\, /\//\/\\, //\\//\\, //\//\\\.
%C Also the number of permutations p of [n] such that in 0p all up-jumps are <= k and no down-jump is larger than 1. An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here. A(4,2) = 9: 1234, 1243, 1324, 1342, 2134, 2143, 2314, 2341, 2431.
%H Alois P. Heinz, <a href="/A288942/b288942.txt">Rows n = 0..140, flattened</a>
%H N. Hein and J. Huang, <a href="http://arxiv.org/abs/1508.01688">Modular Catalan Numbers</a>, arXiv:1508.01688 [math.CO], 2015
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A(n,k) = Sum_{j=0..k} A203717(n,j).
%F G.f. of column k: G(x)/x where G(x) is the reversion of x*(1-x)/(1-x^(k+1)). - _Andrew Howroyd_, Nov 30 2017
%F G.f. g_k(x) of column k satisfies: g_k(x) = Sum_{j=0..k} (x*g_k(x))^j. - _Alois P. Heinz_, May 05 2019
%F A(n,k) = Sum_{j=0..n/(k+1)} (-1)^j * binomial(n+1,j) * binomial(2*n-j*(k+1),n). [Hein Eq (10)] - _R. J. Mathar_, Oct 14 2022
%e A(4,2) = 9:
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%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 2, 2, 2, 2, 2, 2, ...
%e 0, 1, 4, 5, 5, 5, 5, 5, 5, ...
%e 0, 1, 9, 13, 14, 14, 14, 14, 14, ...
%e 0, 1, 21, 36, 41, 42, 42, 42, 42, ...
%e 0, 1, 51, 104, 125, 131, 132, 132, 132, ...
%e 0, 1, 127, 309, 393, 421, 428, 429, 429, ...
%e 0, 1, 323, 939, 1265, 1385, 1421, 1429, 1430, ...
%p b:= proc(u, o, k) option remember; `if`(u+o=0, 1,
%p add(b(u-j, o+j-1, k), j=1..min(1, u))+
%p add(b(u+j-1, o-j, k), j=1..min(k, o)))
%p end:
%p A:= (n, k)-> b(0, n, k):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t b[u_, o_, k_] := b[u, o, k] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1, k], {j, 1, Min[1, u]}] + Sum[b[u + j - 1, o - j, k], {j, 1, Min[k, o]}]];
%t A[n_, k_] := b[0, n, k];
%t Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Oct 27 2017, translated from Maple *)
%o (PARI)
%o T(n,k)=polcoeff(serreverse(x*(1-x)/(1-x*x^k) + O(x^2*x^n)), n+1);
%o for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 29 2017
%Y Columns k=0..10 give: A000007, A000012, A001006, A036765, A036766, A036767, A036768, A036769, A291823, A291824, A291825.
%Y Main diagonal (and upper diagonals) give A000108.
%Y First lower diagonal gives A001453.
%Y Cf. A203717.
%K nonn,tabl
%O 0,13
%A _Alois P. Heinz_, Sep 01 2017
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