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A255704 Number T(n,k) of n-node rooted trees in which the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root equals k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%I A255704 #23 Sep 05 2018 12:33:34
%S A255704 1,0,1,0,1,1,0,2,1,1,0,4,3,1,1,0,8,7,3,1,1,0,17,18,8,3,1,1,0,36,45,21,
%T A255704 8,3,1,1,0,79,116,56,22,8,3,1,1,0,175,298,152,59,22,8,3,1,1,0,395,776,
%U A255704 413,163,60,22,8,3,1,1,0,899,2025,1131,450,166,60,22,8,3,1,1
%N A255704 Number T(n,k) of n-node rooted trees in which the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root equals k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%H A255704 Alois P. Heinz, <a href="/A255704/b255704.txt">Rows n = 1..141, flattened</a>
%F A255704 T(n,1) = A255636(n,1), T(n,k) = A255636(n,k) - A255636(n,k-1) for k>1.
%e A255704 :    o      o     o         o     o     o     o
%e A255704 :  /( )\   /|\   / \       / \    |     |     |
%e A255704 : o o o o o o o o   o     o   o   o     o     o
%e A255704 : |       | |   |  / \   / \     /|\   / \    |
%e A255704 : o       o o   o o   o o   o   o o o o   o   o
%e A255704 :                       |       |     |   |  / \
%e A255704 :                       o       o     o   o o   o
%e A255704 :                                           |
%e A255704 : T(6,3) = 7                                o
%e A255704 Triangle T(n,k) begins:
%e A255704   1;
%e A255704   0,   1;
%e A255704   0,   1,   1;
%e A255704   0,   2,   1,   1;
%e A255704   0,   4,   3,   1,  1;
%e A255704   0,   8,   7,   3,  1,  1;
%e A255704   0,  17,  18,   8,  3,  1, 1;
%e A255704   0,  36,  45,  21,  8,  3, 1, 1;
%e A255704   0,  79, 116,  56, 22,  8, 3, 1, 1;
%e A255704   0, 175, 298, 152, 59, 22, 8, 3, 1, 1;
%p A255704 with(numtheory):
%p A255704 g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
%p A255704       `if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)
%p A255704     end:
%p A255704 T:= (n, k)-> g(n-1, k) -`if`(k=1, 0, g(n-1, k-1)):
%p A255704 seq(seq(T(n, k), k=1..n), n=1..14);
%t A255704 g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[DivisorSum[j, #*(g[#-1, k] - If[# == k, 1, 0])&] * g[n-j, k], {j, 1, n}]/n];
%t A255704 T[n_, k_] := g[n-1, k] - If[k == 1, 0, g[n-1, k-1]];
%t A255704 Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 24 2017, translated from Maple *)
%Y A255704 Columns k=1-10 give: A063524, A002955 (for n>1), A318899, A318900, A318901, A318902, A318903, A318904, A318905, A318906.
%Y A255704 Row sums give A000081.
%Y A255704 T(2*n+1,n+1) gives A255705.
%Y A255704 Cf. A255636.
%K A255704 nonn,tabl
%O A255704 1,8
%A A255704 _Alois P. Heinz_, Mar 02 2015