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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.
4
1, 1, 3, 8, 22, 60, 167, 465, 1306, 3681, 10422, 29597, 84313, 240757, 689035, 1975753, 5675145, 16326198, 47032200, 135658367, 391733593, 1132357784, 3276330780, 9487885056, 27497891241, 79753806451, 231474005120, 672250119756, 1953523496677, 5680002466125
OFFSET
0,3
LINKS
FORMULA
a(n) = A255704(2*n+1,n+1).
a(n) ~ c * d^n / sqrt(n), where d = A051491 = 2.955765285651994974714817524... and c = 0.70755335886284109851526791506579... . - Vaclav Kotesovec, Feb 28 2016
a(n) = A318754(2n+2,n+1) = A318758(2n+2,n+1). - Alois P. Heinz, Sep 02 2018
MAPLE
with(numtheory):
g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
`if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)
end:
a:= a-> g(2*n, n+1) -`if`(n=0, 0, g(2*n, n)):
seq(a(n), n=0..40);
MATHEMATICA
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];
a[n_] := g[2n, n+1] - If[n == 0, 0, g[2n, n]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Mar 02 2015
STATUS
approved