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Evolutionary trees of magnitude n.
(Formerly M3628)
2

%I M3628 #23 Mar 30 2020 04:19:14

%S 1,1,4,28,301,4466,84974,1974904,54233540,1718280152,61695193880,

%T 2475688513024,109797950475448,5333253012414224,281576039542538368,

%U 16055279332196218624,983264280857581866112,64369946360185677026048,4485859513184032011682304,331558482325457407154881024

%N Evolutionary trees of magnitude n.

%D L. R. Foulds and R. W. Robinson, Counting certain classes of evolutionary trees with singleton labels, Congress. Num., 44 (1984), 65-88.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H 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 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

%p Q := proc(n)

%p option remember ;

%p if n <= 1 then

%p 0;

%p else

%p A007151(n)-A007151(n-1) +(n-1)*procname(n-1) ; # eq (3.5)

%p %/2 ;

%p end if;

%p end proc:

%p A007152 := proc(n)

%p if n = 1 then

%p 1;

%p else

%p A007151(n-1)+Q(n-1) ; # eq (3.9)

%p end if ;

%p end proc:

%p seq(A007152(n),n=1..20 ); # _R. J. Mathar_, Mar 19 2018

%t m = 20;

%t A007151 = Rest[Range[0, m]! CoefficientList[ InverseSeries[ Series[(2x - E^x + 1)/(x + 1), {x, 0, m}], x], x]];

%t Q[n_] := Q[n] = If[n <= 1, 0, (1/2)(-A007151[[n - 1]] + A007151[[n]] + (n - 1) Q[n - 1])];

%t a[n_] := If[n == 1, 1, A007151[[n - 1]] + Q[n - 1]];

%t Array[a, m] (* _Jean-François Alcover_, Mar 30 2020, from Maple *)

%Y Cf. A007151.

%K nonn,easy

%O 1,3

%A _Robert W. Robinson_