A038078 Number of identity trees with 2-colored nodes.

1, 2, 1, 2, 6, 20, 69, 270, 1026, 4120, 16794, 70230, 298306, 1288912, 5642559, 25007756, 111998920, 506348902, 2308338456, 10602357346, 49026021552, 228085486580, 1067020210339, 5016982766202, 23698640081356, 112422573858292, 535414026652828, 2559204304109868

Offset: 0

Author

Christian G. Bower, Jan 04 1999

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• Index entries for sequences related to trees

Crossrefs

Cf. A000220. A038077-A038080.

Cf. A246312.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(2*b(i-1$2), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n=0, 1, 2*b(n-1$2) -2*add(b(j-1$2)*b(n-j-1$2)
        , j=1..n-1) -`if`(irem(n, 2, 'r')=0, b(r-1$2), 0)):
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 02 2013

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[2*b[i-1, i-1], j]*b[n-i*j, i-1], {j, 0, n/i}]]];
a[n_] := If[n==0, 1, 2*b[n-1, n-1] - 2*Sum[b[j-1, j-1]*b[n-j-1, n-j-1], {j, 1, n-1}] - If[Mod[n, 2]==0, r=n/2; b[r-1, r-1], 0]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

Formula

G.f.: B(x) - B^2(x)/2 - B(x^2)/2, where B(x) is g.f. for A038077.

a(n) ~ c * d^n / n^(5/2), where d = A246312 = 5.2490324912281705791649522161843092..., c = 0.356142078281568492877259973613... . - Vaclav Kotesovec, Sep 06 2014