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A007152 Evolutionary trees of magnitude n.

%I A007152 M3628 #23 Mar 30 2020 04:19:14
%S A007152 1,1,4,28,301,4466,84974,1974904,54233540,1718280152,61695193880,
%T A007152 2475688513024,109797950475448,5333253012414224,281576039542538368,
%U A007152 16055279332196218624,983264280857581866112,64369946360185677026048,4485859513184032011682304,331558482325457407154881024
%N A007152 Evolutionary trees of magnitude n.
%D A007152 L. R. Foulds and R. W. Robinson, Counting certain classes of evolutionary trees with singleton labels, Congress. Num., 44 (1984), 65-88.
%D A007152 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007152 L. R. Foulds and R. W. Robinson, <a href="/A007151/a007151.pdf">Counting certain classes of evolutionary trees with singleton labels</a>, Congress. Num., 44 (1984), 65-88. (Annotated scanned copy)
%H A007152 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%p A007152 Q  := proc(n)
%p A007152     option remember ;
%p A007152     if n <= 1 then
%p A007152         0;
%p A007152     else
%p A007152         A007151(n)-A007151(n-1) +(n-1)*procname(n-1) ; # eq (3.5)
%p A007152         %/2 ;
%p A007152     end if;
%p A007152 end proc:
%p A007152 A007152 := proc(n)
%p A007152     if n = 1 then
%p A007152         1;
%p A007152     else
%p A007152         A007151(n-1)+Q(n-1) ; # eq (3.9)
%p A007152     end if ;
%p A007152 end proc:
%p A007152 seq(A007152(n),n=1..20 ); # _R. J. Mathar_, Mar 19 2018
%t A007152 m = 20;
%t A007152 A007151 = Rest[Range[0, m]! CoefficientList[ InverseSeries[ Series[(2x - E^x + 1)/(x + 1), {x, 0, m}], x], x]];
%t A007152 Q[n_] := Q[n] = If[n <= 1, 0, (1/2)(-A007151[[n - 1]] + A007151[[n]] + (n - 1) Q[n - 1])];
%t A007152 a[n_] := If[n == 1, 1, A007151[[n - 1]] + Q[n - 1]];
%t A007152 Array[a, m] (* _Jean-François Alcover_, Mar 30 2020, from Maple *)
%Y A007152 Cf. A007151.
%K A007152 nonn,easy
%O A007152 1,3
%A A007152 _Robert W. Robinson_