A000675 Number of centered 3-valent (or boron, or binary) trees with n nodes. (Formerly M0977 N0366)

1, 1, 0, 1, 1, 1, 2, 4, 5, 10, 19, 36, 68, 138, 277, 581, 1218, 2591, 5545, 12026, 26226, 57719, 127685, 284109, 634919, 1425516, 3212890, 7269605, 16504439, 37592604, 85876345, 196717882, 451768247, 1039990913, 2399476030, 5547849750

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.

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;
S3[f_, h_, x_] := f[h, x]^3/6 + f[h, x] f[h, x^2]/2 + f[h, x^3]/3;
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) + S3[T, h-1, z]z - S3[T, h-2, z]z - (T[h-1, z] - T[h-2, z]) (T[h-1, z]-1), z], n+1], {h, 1, n/2}] + PadRight[{1, 1}, n+1] (* Robert A. Russell, Sep 15 2018 *)

Crossrefs

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

Sequence in context: A018424 A240100 A326156 * A005018 A249399 A118551

Adjacent sequences: A000672 A000673 A000674 * A000676 A000677 A000678

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved