A000673 Number of bicentered 3-valent (or boron, or binary) trees with n nodes. (Formerly M0355 N0133)

0, 0, 1, 0, 1, 1, 2, 2, 6, 8, 18, 30, 67, 127, 275, 551, 1192, 2507, 5475, 11820, 26007, 57077, 126686, 281625, 630660, 1416116, 3195784, 7232624, 16430563, 37429146, 85528079, 195940960, 450074270, 1036226173, 2391193488, 5529420585

Offset

0,7

References

A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 451).

R. C. Read, personal communication.

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

Table of n, a(n) for n=0..35.

Nicolas Broutin and Philippe Flajolet, The distribution of height and diameter in random non-plane binary trees, Random Struct. Algorithms 41, No. 2, 215-252 (2012).

E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.

R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976

Index entries for sequences related to trees

Mathematica

n = 50; (* algorithm from Rains and Sloane *)
S2[f_, h_, x_] := f[h, x]^2/2 + f[h, x^2]/2;
T[-1, z_] := 1;  T[h_, z_] := T[h, z] = Table[z^k, {k, 0, n}].Take[CoefficientList[z^(n+1) + 1 + S2[T, h-1, z]z, z], n+1];
Sum[Take[CoefficientList[z^(n+1) + (T[h, z] - T[h-1, z])^2/2 + (T[h, z^2] - T[h-1, z^2])/2, z], n+1], {h, 0, n/2}] (* Robert A. Russell, Sep 15 2018 *)

Crossrefs

A000672 = A000673 + A000675. Cf. A000022, A000200, A000602.

Sequence in context: A214932 A322132 A054153 * A355640 A370586 A276425

Adjacent sequences: A000670 A000671 A000672 * A000674 A000675 A000676

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved