A000251 -id:A000251 - cp's OEIS frontend

A034853 Triangle giving number of trees with n >= 3 nodes and diameter d >= 2.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 5, 2, 1, 1, 3, 8, 7, 3, 1, 1, 3, 14, 14, 11, 3, 1, 1, 4, 21, 32, 29, 14, 4, 1, 1, 4, 32, 58, 74, 42, 19, 4, 1, 1, 5, 45, 110, 167, 128, 66, 23, 5, 1

Offset: 3

N. J. A. Sloane

			   1
   1    1
   1    1    1
   1    2    2    1
   1    2    5    2    1
   1    3    8    7    3    1
   1    3   14   14   11    3    1
   1    4   21   32   29   14    4    1
   1    4   32   58   74   42   19    4    1
   1    5   45  110  167  128   66   23    5    1
   1    5   65  187  367  334  219   88   29    5    1
   1    6   88  322  755  850  645  328  123   34    6    1

R. J. Mathar, Table of n, a(n) for n = 3..212 (a(192) corrected by _Sean A. Irvine_, Apr 28 2022)
B. D. McKay, Lists of Trees sorted by diameter and Homeomorphically irreducible trees, with <= 22 nodes.
B. D. McKay, Lists of Trees sorted by diameter and Homeomorphically irreducible trees, with <= 22 nodes. [Cached copy of top page only, pdf file, no active links, with permission]
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Index entries for sequences related to trees

Cf. A000055 (row sums), A283826, A000094 (diameter 4), A000147 (diameter 5), A000251 (diameter 6), A000550 (diameter 7), A000306 (diameter 8).

Reference gives recurrence.

A000147 Number of trees of diameter 5.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

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

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).

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.
Index entries for sequences related to trees

Cf. A034853, A000251 (diameter 6).

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

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 *)

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

More terms from Franklin T. Adams-Watters, Jan 13 2006

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A034853 Triangle giving number of trees with n >= 3 nodes and diameter d >= 2.

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A000147 Number of trees of diameter 5.

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