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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.

Index entries for sequences related to trees

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-Fran├žois Alcover, Feb 17 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A034853, A000251 (diameter 6).

Sequence in context: A034791 A140253 A018453 * A128902 A227213 A060552

Adjacent sequences:  A000144 A000145 A000146 * A000148 A000149 A000150

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified May 25 01:01 EDT 2017. Contains 287008 sequences.