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 A000147 Number of trees of diameter 5. (Formerly M1741 N0690) 3
 0, 0, 0, 0, 0, 1, 2, 7, 14, 32, 58, 110, 187, 322, 519, 839, 1302, 2015, 3032, 4542, 6668, 9738, 14006, 20036, 28324, 39830, 55473, 76875, 105692, 144629, 196585, 266038, 357952, 479664, 639519, 849425, 1123191, 1479972, 1942284, 2540674, 3311415 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS 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 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478. FORMULA 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 MAPLE b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0,      add(binomial(b((i-1)\$2, k-1)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))     end: g:= n-> b((n-1)\$2, 2) -b((n-1)\$2, 1): a:= n-> (add(g(i)*g(n-i), i=0..n)+`if`(n::even, g(n/2), 0))/2: seq(a(n), n=1..45);  # Alois P. Heinz, Feb 09 2016 MATHEMATICA 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 *) CROSSREFS Cf. A034853, A000251 (diameter 6). Sequence in context: A286829 A286861 A290682 * A128902 A227213 A319455 Adjacent sequences:  A000144 A000145 A000146 * A000148 A000149 A000150 KEYWORD nonn AUTHOR EXTENSIONS More terms from Franklin T. Adams-Watters, Jan 13 2006 STATUS approved

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Last modified December 6 04:14 EST 2019. Contains 329784 sequences. (Running on oeis4.)