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A038076 Number of rooted identity trees with 3-colored leaves.

%I A038076 #24 May 20 2023 15:11:49
%S A038076 3,3,6,16,46,142,461,1542,5278,18417,65218,233816,846938,3094943,
%T A038076 11395715,42237936,157465847,590075550,2221391912,8397223487,
%U A038076 31861406058,121300625969,463233477550,1774034788166,6811612470692,26216538077715,101125406981562
%N A038076 Number of rooted identity trees with 3-colored leaves.
%H A038076 Alois P. Heinz, <a href="/A038076/b038076.txt">Table of n, a(n) for n = 1..600</a>
%H A038076 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A038076 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A038076 Shifts left under Weigh transform.
%F A038076 a(n) ~ c * d^n / n^(3/2), where d = 4.0814589930714884560076189705..., c = 0.4583632659157592121544633778... . - _Vaclav Kotesovec_, Sep 06 2014
%F A038076 G.f. A(x) satisfies: A(x) = 2*x + x * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k) / k ). - _Ilya Gutkovskiy_, May 19 2023
%p A038076 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A038076       add(binomial(a(i$2), j)*b(n-i*j, i-1), j=0..n/i)))
%p A038076     end:
%p A038076 a:= n-> `if`(n<2, 3*n, b(n-1, n-1)):
%p A038076 seq(a(n), n=1..35);  # _Alois P. Heinz_, Aug 01 2013
%t A038076 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[a[i], j]*b[n - i*j, i-1], {j, 0, n/i}]]];
%t A038076 a[n_] := If[n<2, 3*n, b[n-1, n-1]];
%t A038076 Table[a[n], {n, 1, 35}] (* _Jean-François Alcover_, Mar 01 2016, after _Alois P. Heinz_ *)
%Y A038076 Cf. A004111, A038075.
%K A038076 nonn
%O A038076 1,1
%A A038076 _Christian G. Bower_, Jan 04 1999