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A288942 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.

%I A288942 #60 Jul 07 2024 21:17:58
%S A288942 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 A288942 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 A288942 125,309,323,1,0,1,1,2,5,14,42,131,393,939,835,1,0
%N A288942 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 A288942 Also the number of Dyck paths of semilength n with all ascent lengths <= k. A(4,2) = 9: /\/\/\/\, //\\/\/\, /\//\\/\, /\/\//\\, //\/\\/\, //\/\/\\, /\//\/\\, //\\//\\, //\//\\\.
%C A288942 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 A288942 Alois P. Heinz, <a href="/A288942/b288942.txt">Rows n = 0..140, flattened</a>
%H A288942 N. Hein and J. Huang, <a href="http://arxiv.org/abs/1508.01688">Modular Catalan Numbers</a>, arXiv:1508.01688 [math.CO], 2015
%H A288942 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A288942 A(n,k) = Sum_{j=0..k} A203717(n,j).
%F A288942 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 A288942 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 A288942 A(n,k) = Sum_{j=0..n/(k+1)} (-1)^j/(n+1) * binomial(n+1,j) * binomial(2*n-j*(k+1),n). [Hein Eq (10)] - _R. J. Mathar_, Oct 14 2022; corrected by _Tijn Caspar de Leeuw_, Jul 07 2024
%e A288942 A(4,2) = 9:
%e A288942 .
%e A288942 .   o    o      o      o      o      o       o      o       o
%e A288942 .   |    |      |      |     / \    / \     / \    / \     / \
%e A288942 .   o    o      o      o    o   o  o   o   o   o  o   o   o   o
%e A288942 .   |    |     / \    / \   |          |  ( )        ( )  |   |
%e A288942 .   o    o    o   o  o   o  o          o  o o        o o  o   o
%e A288942 .   |   / \   |          |  |          |
%e A288942 .   o  o   o  o          o  o          o
%e A288942 .   |
%e A288942 .   o
%e A288942 .
%e A288942 Square array A(n,k) begins:
%e A288942   1, 1,   1,   1,    1,    1,    1,    1,    1, ...
%e A288942   0, 1,   1,   1,    1,    1,    1,    1,    1, ...
%e A288942   0, 1,   2,   2,    2,    2,    2,    2,    2, ...
%e A288942   0, 1,   4,   5,    5,    5,    5,    5,    5, ...
%e A288942   0, 1,   9,  13,   14,   14,   14,   14,   14, ...
%e A288942   0, 1,  21,  36,   41,   42,   42,   42,   42, ...
%e A288942   0, 1,  51, 104,  125,  131,  132,  132,  132, ...
%e A288942   0, 1, 127, 309,  393,  421,  428,  429,  429, ...
%e A288942   0, 1, 323, 939, 1265, 1385, 1421, 1429, 1430, ...
%p A288942 b:= proc(u, o, k) option remember; `if`(u+o=0, 1,
%p A288942       add(b(u-j, o+j-1, k), j=1..min(1, u))+
%p A288942       add(b(u+j-1, o-j, k), j=1..min(k, o)))
%p A288942     end:
%p A288942 A:= (n, k)-> b(0, n, k):
%p A288942 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A288942 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 A288942 A[n_, k_] := b[0, n, k];
%t A288942 Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Oct 27 2017, translated from Maple *)
%o A288942 (PARI)
%o A288942 T(n,k)=polcoeff(serreverse(x*(1-x)/(1-x*x^k) + O(x^2*x^n)), n+1);
%o A288942 for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 29 2017
%Y A288942 Columns k=0..10 give: A000007, A000012, A001006, A036765, A036766, A036767, A036768, A036769, A291823, A291824, A291825.
%Y A288942 Main diagonal (and upper diagonals) give A000108.
%Y A288942 First lower diagonal gives A001453.
%Y A288942 Cf. A203717.
%K A288942 nonn,tabl
%O A288942 0,13
%A A288942 _Alois P. Heinz_, Sep 01 2017