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%I #33 Feb 28 2022 08:14:28
%S 2,2,5,13,37,108,332,1042,3360,11019,36722,123875,422449,1453553,
%T 5040816,17599468,61814275,218252584,774226549,2758043727,9862357697,
%U 35387662266,127374191687,459783039109,1664042970924,6037070913558,21951214425140,79981665585029
%N Number of rooted trees with 2-colored leaves.
%H Alois P. Heinz, <a href="/A029856/b029856.txt">Table of n, a(n) for n = 1..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=768">Encyclopedia of Combinatorial Structures 768</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 G.f. satisfies: A(x) = x + x*exp( Sum_{n>=1} A(x^n)/n ). - _Paul D. Hanna_, Oct 19 2005
%F a(n) ~ c * d^n / n^(3/2), where d = 3.848442876944251389076286931217197... and c = 0.48335853985605895591573724406549734... - _Vaclav Kotesovec_, Mar 29 2014
%p A:= proc(n) option remember; if n=0 then 0 else convert(series(x+x* exp(sum(subs(x=x^i, A(n-1))/i, i=1..n-1)), x=0, n+1), polynom) fi end; a:= n-> coeff(A(n), x,n): seq(a(n), n=1..25); # _Alois P. Heinz_, Aug 22 2008
%p # second Maple program:
%p with(numtheory): a:= proc(n) option remember; local d,j; if n<=1 then 2*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..25); # _Alois P. Heinz_, Sep 06 2008
%t a[n_] := a[n] = If [n <= 1, 2*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)]; Array[a, 25] (* _Jean-François Alcover_, Mar 13 2015, after _Alois P. Heinz_ *)
%o (PARI) {a(n)=local(A=x+x*O(x^n));for(i=1,n, A=x+x*exp(sum(m=1,n,subst(A,x,x^m)/m)));polcoeff(A,n,x)} \\ _Paul D. Hanna_, Oct 19 2005
%Y Essentially the same as A036249.
%Y Cf. A000081, A029857, A038049.
%K nonn,easy,eigen
%O 1,1
%A _Christian G. Bower_
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