A000550 Number of trees of diameter 7.

1, 3, 14, 42, 128, 334, 850, 2010, 4625, 10201, 21990, 46108, 94912, 191562, 380933, 746338, 1444676, 2763931, 5235309, 9822686, 18275648, 33734658, 61826344, 112550305, 203627610, 366267931, 655261559, 1166312530, 2066048261, 3643352362, 6397485909, 11188129665, 19491131627, 33831897511, 58519577756, 100885389220, 173368983090, 297021470421, 507378371670, 864277569606, 1468245046383, 2487774321958, 4204663810414, 7089200255686, 11924621337321, 20012746962064, 33513139512868, 56001473574091, 93387290773141, 155419866337746

Offset: 8

Author

N. J. A. Sloane

Keywords

• nonn

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 = 8..2500
• J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
• Index entries for sequences related to trees

Crossrefs

Cf. A034853, A000306 (diameter 8)

Programs

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, 3) -b((n-1)$2, 2):
a:= n-> (add(g(i)*g(n-i), i=0..n)+`if`(n::even, g(n/2), 0))/2:
seq(a(n), n=8..40);  # Alois P. Heinz, Feb 09 2016

Mathematica

m = 50; r[x_] = (Rest @ CoefficientList[ Series[ x*Product[ (1 - x^k)^(- PartitionsP[k-1]), {k, 1, m+3}], {x, 0, m+3}], x] - PartitionsP[ Range[0, m+2]]).(x^Range[m+3]); A000550 = CoefficientList[(r[x]^2 + r[x^2])/2, x][[9 ;; m+8]] (* Jean-François Alcover, Feb 09 2016 *)

Formula

G.f.: a(x)=(r(x)^2+r(x^2))/2, where r(x) is the generating function of A000235. - Sean A. Irvine, Nov 21 2010

Extensions

More terms from Sean A. Irvine, Nov 21 2010