A029857 Number of rooted trees with 3-colored leaves.

3, 3, 9, 28, 94, 328, 1197, 4486, 17235, 67429, 267932, 1078003, 4383784, 17987897, 74385984, 309694232, 1297037177, 5460726214, 23098296648, 98113995068, 418335662448, 1789814398035, 7681522429474, 33061825858259, 142674028869587, 617180102839217

Offset

1,1

Links

Alois P. Heinz, Table of n, a(n) for n = 1..500

N. J. A. Sloane, Transforms

Index entries for sequences related to rooted trees

Formula

Shifts left under Euler transform.

a(n) ~ c * d^n / n^(3/2), where d = 4.58859196701042554480382685... and c = 0.5102557157321640697473838... - Vaclav Kotesovec, Mar 29 2014

G.f. A(x) satisfies: A(x) = 2*x + x * exp( Sum_{k>=1} A(x^k) / k ). - Ilya Gutkovskiy, May 19 2023

Maple

with(numtheory): a:= proc(n) option remember; local d, j; if n<=1 then 3*n else (add(d*a(d), d=divisors(n-1)) +add(add(d*a(d), d=divisors(j)) *a(n-j), j=1..n-2))/ (n-1) fi end: seq(a(n), n=1..30); # Alois P. Heinz, Sep 06 2008

Mathematica

a[n_] := a[n] = If[n<=1, 3*n, (Sum[d*a[d], {d, Divisors[n-1]}] + Sum[Sum[ d*a[d], {d, Divisors[j]}]*a[n-j], {j, 1, n-2}])/(n-1)]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 21 2016 *)

Crossrefs

Cf. A000081, A029856, A038050.

Sequence in context: A229024 A117976 A010098 * A327712 A257611 A268617

Adjacent sequences: A029854 A029855 A029856 * A029858 A029859 A029860

Keyword

nonn,easy,eigen

Author

Christian G. Bower

Status

approved