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A038078 Number of identity trees with 2-colored nodes.

%I A038078 #22 Mar 01 2016 04:45:43
%S A038078 1,2,1,2,6,20,69,270,1026,4120,16794,70230,298306,1288912,5642559,
%T A038078 25007756,111998920,506348902,2308338456,10602357346,49026021552,
%U A038078 228085486580,1067020210339,5016982766202,23698640081356,112422573858292,535414026652828,2559204304109868
%N A038078 Number of identity trees with 2-colored nodes.
%H A038078 Alois P. Heinz, <a href="/A038078/b038078.txt">Table of n, a(n) for n = 0..500</a>
%H A038078 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A038078 G.f.: B(x) - B^2(x)/2 - B(x^2)/2, where B(x) is g.f. for A038077.
%F A038078 a(n) ~ c * d^n / n^(5/2), where d = A246312 = 5.2490324912281705791649522161843092..., c = 0.356142078281568492877259973613... . - _Vaclav Kotesovec_, Sep 06 2014
%p A038078 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A038078       add(binomial(2*b(i-1$2), j)*b(n-i*j, i-1), j=0..n/i)))
%p A038078     end:
%p A038078 a:= n-> `if`(n=0, 1, 2*b(n-1$2) -2*add(b(j-1$2)*b(n-j-1$2)
%p A038078         , j=1..n-1) -`if`(irem(n, 2, 'r')=0, b(r-1$2), 0)):
%p A038078 seq(a(n), n=0..35);  # _Alois P. Heinz_, Aug 02 2013
%t A038078 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}]]];
%t A038078 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]];
%t A038078 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Mar 01 2016, after _Alois P. Heinz_ *)
%Y A038078 Cf. A000220. A038077-A038080.
%Y A038078 Cf. A246312.
%K A038078 nonn
%O A038078 0,2
%A A038078 _Christian G. Bower_, Jan 04 1999