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%I M1741 N0690 #32 Apr 03 2017 04:22:36
%S 0,0,0,0,0,1,2,7,14,32,58,110,187,322,519,839,1302,2015,3032,4542,
%T 6668,9738,14006,20036,28324,39830,55473,76875,105692,144629,196585,
%U 266038,357952,479664,639519,849425,1123191,1479972,1942284,2540674,3311415
%N Number of trees of diameter 5.
%C A tree of diameter 5 is formed from two rooted trees of height 2, with their roots joined. - _Franklin T. Adams-Watters_, Jan 13 2006
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A000147/b000147.txt">Table of n, a(n) for n = 1..1000</a>
%H 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 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F If n odd, a(n) = Sum_{k=1..(n-1)/2} b(k)*b(n-k); if n even, a(n) = (Sum_{k=1..n/2-1} b(k)*b(n-k)) + C(b(n/2)+1, 2), where b(n) = P(n-1)-1 = A000065(n-1). - _Franklin T. Adams-Watters_, Jan 13 2006
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0,
%p add(binomial(b((i-1)$2, k-1)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
%p end:
%p g:= n-> b((n-1)$2, 2) -b((n-1)$2, 1):
%p a:= n-> (add(g(i)*g(n-i), i=0..n)+`if`(n::even, g(n/2), 0))/2:
%p seq(a(n), n=1..45); # _Alois P. Heinz_, Feb 09 2016
%t b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1 || k<1, 0, Sum[Binomial[ b[i-1, i-1, k-1]+j-1, j]*b[n-i*j, i-1, k], {j, 0, n/i}]]]; g[n_] := b[n-1, n-1, 2] - b[n-1, n-1, 1]; a[n_] := (Sum[g[i]*g[n-i], {i, 0, n}] + If[EvenQ[n], g[n/2], 0])/2; Table[a[n], {n, 1, 45}] (* _Jean-François Alcover_, Feb 17 2016, after _Alois P. Heinz_ *)
%Y Cf. A034853, A000251 (diameter 6).
%K nonn
%O 1,7
%A _N. J. A. Sloane_
%E More terms from _Franklin T. Adams-Watters_, Jan 13 2006
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