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A244530 Number T(n,k) of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

%I A244530 #22 Sep 07 2018 21:12:11
%S A244530 1,0,1,0,1,1,0,4,0,1,0,11,2,0,1,0,36,5,0,0,1,0,117,11,3,0,0,1,0,393,
%T A244530 28,7,0,0,0,1,0,1339,78,8,4,0,0,0,1,0,4630,201,21,9,0,0,0,0,1,0,16193,
%U A244530 532,55,10,5,0,0,0,0,1,0,57201,1441,121,11,11,0,0,0,0,0,1
%N A244530 Number T(n,k) of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.
%C A244530 T(1,0) = 1 by convention.
%H A244530 Alois P. Heinz, <a href="/A244530/b244530.txt">Rows n = 1..141, flattened</a>
%e A244530 T(5,1) = 11:
%e A244530   o   o     o     o     o     o     o     o     o     o     o
%e A244530   |   |     |     |    / \   / \   / \    |    /|\   /|\   /|\
%e A244530   o   o     o     o   o   o o   o o   o   o   o o o o o o o o o
%e A244530   |   |    / \   / \  |         | |   |  /|\  |       |       |
%e A244530   o   o   o   o o   o o         o o   o o o o o       o       o
%e A244530   |  / \  |         | |         |
%e A244530   o o   o o         o o         o
%e A244530   |
%e A244530   o
%e A244530 Triangle T(n,k) begins:
%e A244530   1;
%e A244530   0,     1;
%e A244530   0,     1,   1;
%e A244530   0,     4,   0,  1;
%e A244530   0,    11,   2,  0,  1;
%e A244530   0,    36,   5,  0,  0, 1;
%e A244530   0,   117,  11,  3,  0, 0, 1;
%e A244530   0,   393,  28,  7,  0, 0, 0, 1;
%e A244530   0,  1339,  78,  8,  4, 0, 0, 0, 1;
%e A244530   0,  4630, 201, 21,  9, 0, 0, 0, 0, 1;
%e A244530   0, 16193, 532, 55, 10, 5, 0, 0, 0, 0, 1;
%p A244530 b:= proc(n, t, k) option remember; `if`(n=0,
%p A244530       `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
%p A244530        b(n-j, max(0, t-1), k), j=1..n)))
%p A244530     end:
%p A244530 T:= (n, k)-> b(n-1, k$2) -`if`(n=1 and k=0, 0, b(n-1, k+1$2)):
%p A244530 seq(seq(T(n, k), k=0..n-1), n=1..14);
%t A244530 b[n_, t_, k_] := b[n, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[t>n, 0, Sum[b[j-1, k, k]*b[n-j, Max[0, t-1], k], {j, 1, n}]]]; T[n_, k_] := b[n-1, k, k] - If[n == 1 && k == 0, 0, b[n-1, k+1, k+1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Jan 13 2015, translated from Maple *)
%Y A244530 Columns k=0-10 give: A063524, A106640(n-2), A244531, A244532, A244533, A244534, A244535, A244536, A244537, A244538, A244539.
%Y A244530 Row sums give A000108(n-1).
%Y A244530 Cf. A244454 (unordered unlabeled rooted trees).
%K A244530 nonn,tabl
%O A244530 1,8
%A A244530 _Joerg Arndt_ and _Alois P. Heinz_, Jun 29 2014