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A255705 Number of 2n+1-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 n+1.

%I A255705 #16 Sep 03 2018 20:18:57
%S A255705 1,1,3,8,22,60,167,465,1306,3681,10422,29597,84313,240757,689035,
%T A255705 1975753,5675145,16326198,47032200,135658367,391733593,1132357784,
%U A255705 3276330780,9487885056,27497891241,79753806451,231474005120,672250119756,1953523496677,5680002466125
%N A255705 Number of 2n+1-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 n+1.
%H A255705 Alois P. Heinz, <a href="/A255705/b255705.txt">Table of n, a(n) for n = 0..800</a>
%F A255705 a(n) = A255704(2*n+1,n+1).
%F A255705 a(n) ~ c * d^n / sqrt(n), where d = A051491 = 2.955765285651994974714817524... and c = 0.70755335886284109851526791506579... . - _Vaclav Kotesovec_, Feb 28 2016
%F A255705 a(n) = A318754(2n+2,n+1) = A318758(2n+2,n+1). - _Alois P. Heinz_, Sep 02 2018
%p A255705 with(numtheory):
%p A255705 g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
%p A255705       `if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)
%p A255705     end:
%p A255705 a:= a-> g(2*n, n+1) -`if`(n=0, 0, g(2*n, n)):
%p A255705 seq(a(n), n=0..40);
%t A255705 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 A255705 a[n_] :=  g[2n, n+1] - If[n == 0, 0, g[2n, n]];
%t A255705 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Mar 24 2017, translated from Maple *)
%Y A255705 Cf. A255636, A255704, A318754, A318758.
%K A255705 nonn
%O A255705 0,3
%A A255705 _Alois P. Heinz_, Mar 02 2015