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A000550 Number of trees of diameter 7.

%I A000550 M2969 N1201 #34 Apr 03 2017 03:36:54
%S A000550 1,3,14,42,128,334,850,2010,4625,10201,21990,46108,94912,191562,
%T A000550 380933,746338,1444676,2763931,5235309,9822686,18275648,33734658,
%U A000550 61826344,112550305,203627610,366267931,655261559,1166312530,2066048261,3643352362,6397485909,11188129665,19491131627,33831897511,58519577756,100885389220,173368983090,297021470421,507378371670,864277569606,1468245046383,2487774321958,4204663810414,7089200255686,11924621337321,20012746962064,33513139512868,56001473574091,93387290773141,155419866337746
%N A000550 Number of trees of diameter 7.
%D A000550 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000550 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000550 Alois P. Heinz, <a href="/A000550/b000550.txt">Table of n, a(n) for n = 8..2500</a>
%H A000550 J. Riordan, <a href="http://dx.doi.org/10.1147/rd.45.0473">Enumeration of trees by height and diameter</a>, IBM J. Res. Dev. 4 (1960), 473-478.
%H A000550 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A000550 G.f.: a(x)=(r(x)^2+r(x^2))/2, where r(x) is the generating function of A000235. - _Sean A. Irvine_, Nov 21 2010
%p A000550 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0,
%p A000550      add(binomial(b((i-1)$2, k-1)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
%p A000550     end:
%p A000550 g:= n-> b((n-1)$2, 3) -b((n-1)$2, 2):
%p A000550 a:= n-> (add(g(i)*g(n-i), i=0..n)+`if`(n::even, g(n/2), 0))/2:
%p A000550 seq(a(n), n=8..40);  # _Alois P. Heinz_, Feb 09 2016
%t A000550 m = 50; r[x_] = (Rest @ CoefficientList[ Series[ x*Product[ (1 - x^k)^(- PartitionsP[k-1]), {k, 1, m+3}], {x, 0, m+3}], x] - PartitionsP[ Range[0, m+2]]).(x^Range[m+3]); A000550 = CoefficientList[(r[x]^2 + r[x^2])/2, x][[9 ;; m+8]] (* _Jean-François Alcover_, Feb 09 2016 *)
%Y A000550 Cf. A034853, A000306 (diameter 8)
%K A000550 nonn
%O A000550 8,2
%A A000550 _N. J. A. Sloane_
%E A000550 More terms from _Sean A. Irvine_, Nov 21 2010