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%I #26 May 20 2023 15:11:41
%S 3,3,9,28,94,328,1197,4486,17235,67429,267932,1078003,4383784,
%T 17987897,74385984,309694232,1297037177,5460726214,23098296648,
%U 98113995068,418335662448,1789814398035,7681522429474,33061825858259,142674028869587,617180102839217
%N Number of rooted trees with 3-colored leaves.
%H Alois P. Heinz, <a href="/A029857/b029857.txt">Table of n, a(n) for n = 1..500</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F Shifts left under Euler transform.
%F a(n) ~ c * d^n / n^(3/2), where d = 4.58859196701042554480382685... and c = 0.5102557157321640697473838... - _Vaclav Kotesovec_, Mar 29 2014
%F G.f. A(x) satisfies: A(x) = 2*x + x * exp( Sum_{k>=1} A(x^k) / k ). - _Ilya Gutkovskiy_, May 19 2023
%p 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
%t 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 *)
%Y Cf. A000081, A029856, A038050.
%K nonn,easy,eigen
%O 1,1
%A _Christian G. Bower_
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