login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000147 Number of trees of diameter 5.
(Formerly M1741 N0690)
4
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
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 * A334069 A128902 A227213
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Franklin T. Adams-Watters, Jan 13 2006
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)