A108530 - OEIS

%I #20 Mar 03 2016 12:12:17

%S 1,1,2,4,12,34,110,364,1248,4356,15520,56022,204726,755472,2812004,

%T 10543718,39791070,151022006,576090250,2207493080,8493196536,

%U 32797115398,127071214442,493831241234,1924504466246,7519182311366,29447430754182,115577336981932

%N Number of rooted identity trees with n internal (non-leaf) nodes.

%C Also for n>0, rooted trees with n nodes and 2-colored internal nodes. Black nodes correspond to nodes with a leaf child; white nodes correspond to those without one.

%H Alois P. Heinz, <a href="/A108530/b108530.txt">Table of n, a(n) for n = 0..1000</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 and halves under WEIGH transform.

%F a(n) ~ c * d^n / n^(3/2), where d = 4.1516890102085520777311008746639624... and c = 0.3329810927479684511418598248... - _Vaclav Kotesovec_, Feb 28 2014

%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, 1, 2*b(n-1,n-1)):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, May 20 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, 1, 2*b[n-1, n-1]];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 01 2016, after _Alois P. Heinz_ *)

%Y Cf. A004111, A004113, A108531, A108532.

%K nonn,eigen

%O 0,3

%A _Christian G. Bower_, Jun 07 2005