A038075 - OEIS

%I #23 May 19 2023 10:57:11

%S 2,2,3,7,16,41,110,304,858,2474,7234,21418,64057,193277,587531,

%T 1797817,5532916,17115442,53186682,165958893,519764706,1633331926,

%U 5148420607,16273962742,51574291758,163834983761,521597902077,1663993969029,5318540288800,17029516243797

%N Number of rooted identity trees with 2-colored leaves.

%H Alois P. Heinz, <a href="/A038075/b038075.txt">Table of n, a(n) for n = 1..600</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 Weigh transform.

%F a(n) ~ c * d^n / n^(3/2), where d = 3.3683668081969694736300401764..., c = 0.4229796097587478606873477... . - _Vaclav Kotesovec_, Sep 10 2014

%F G.f. A(x) satisfies: A(x) = x + x * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k) / k ). - _Ilya Gutkovskiy_, May 19 2023

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add(binomial(a(i$2), j)*b(n-i*j, i-1), j=0..n/i)))

%p     end:

%p a:= n-> `if`(n<2, 2*n, b(n-1, n-1)):

%p seq(a(n), n=1..35);  # _Alois P. Heinz_, Aug 01 2013

%t 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 a[n_] := If[n<2, 2*n, b[n-1, n-1]];

%t Table[a[n], {n, 1, 35}] (* _Jean-François Alcover_, Mar 01 2016, after _Alois P. Heinz_ *)

%Y Cf. A004111, A038076.

%K nonn

%O 1,1

%A _Christian G. Bower_, Jan 04 1999