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
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
Programs
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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
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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 *)
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