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A006965 Number of directed trees with n nodes.

%I A006965 M1677 #34 Nov 01 2020 07:45:28
%S A006965 1,2,6,25,114,591,3298,19532,120687,771373,5061741,33943662,231751331,
%T A006965 1606587482,11283944502,80157645245,575105238243,4162624144308,
%U A006965 30365913761136,223075674659696,1649166676341180,12262121068089094,91649977839972636,688288656744067230
%N A006965 Number of directed trees with n nodes.
%D A006965 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006965 Alois P. Heinz, <a href="/A006965/b006965.txt">Table of n, a(n) for n = 1..1093</a>
%H A006965 P. Leroux and B. Miloudi, <a href="http://www.labmath.uqam.ca/~annales/volumes/16-1/PDF/053-080.pdf">Généralisations de la formule d'Otter</a>, Ann. Sci. Math. Quebec, 1992, Vol. 16, No. 1, 53-80.
%H A006965 P. Leroux and B. Miloudi, <a href="/A000081/a000081_2.pdf">Généralisations de la formule d'Otter</a>, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
%H A006965 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%p A006965 with(combstruct):B:=x->add(3*count([S, {B = Set(S), S = Prod(B,B,B,Z)}, unlabeled],size=i)*x^i,i=1..50);seq(coeff(B(x)-B(x)^2/2+B(x^2)/2,x,n)/3,n=1..30);# with Algolib (Pab Ter)
%p A006965 # second Maple program:
%p A006965 b:= proc(n) option remember; `if`(n<2, 3*n, (add(add(b(d)
%p A006965       *d, d=numtheory[divisors](j))*b(n-j), j=1..n-1))/(n-1))
%p A006965     end:
%p A006965 a:= n-> `if`(n=0, 1, b(n)-(add(b(k) *b(n-k), k=0..n)-
%p A006965         `if`(irem(n, 2)=0, b(n/2), 0))/2)/3:
%p A006965 seq(a(n), n=1..30);  # _Alois P. Heinz_, Jun 03 2020
%t A006965 b[n_] := b[n] = If[n < 2, 3 n, (Sum[Sum[b[d] d, {d, Divisors[j]}] b[n - j], {j, 1, n - 1}])/(n - 1)];
%t A006965 a[n_] := If[n == 0, 1, b[n] - (Sum[b[k] b[n - k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2]/3;
%t A006965 Array[a, 30] (* _Jean-François Alcover_, Nov 01 2020, after _Alois P. Heinz_ *)
%Y A006965 Equals (1/3) A038060(n).
%Y A006965 Row sums of A335362.
%K A006965 nonn
%O A006965 1,2
%A A006965 _Simon Plouffe_