A000269 Number of trees with n nodes, 3 of which are labeled. (Formerly M3014 N1220)

3, 16, 67, 251, 888, 3023, 10038, 32722, 105228, 334836, 1056611, 3311784, 10322791, 32026810, 98974177, 304835956, 936147219, 2867586542, 8764280567, 26733395986, 81399821915, 247459136331, 751211286356, 2277496842016

Offset

3,1

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.

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

T. D. Noe, Table of n, a(n) for n=3..200

Index entries for sequences related to trees

Formula

G.f.: A(x) = B(x)^3*(3-2*B(x))/(1-B(x))^3, where B(x) is g.f. for rooted trees with n nodes, cf. A000081. - Vladeta Jovovic, Oct 19 2001

a(n) = A000524(n) - 2*A000243(n).

Mathematica

b[n_] := b[n] = If[n <= 1, n, Sum[k*b[k]*s[n-1, k], {k, 1, n-1}]/(n-1)]; s[n_, k_] := s[n, k] = Sum[b[n+1 - j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum[ b[k]*x^k, {k, 1, n}]; a[n_] := SeriesCoefficient[ B[n-1]^3 * (2*B[n-1]-3) / (B[n-1]-1)^3, {x, 0, n}]; Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Jan 27 2015 *)

Crossrefs

Column k=3 of A034799.

Sequence in context: A179600 A278089 A248016 * A378406 A370248 A370274

Adjacent sequences: A000266 A000267 A000268 * A000270 A000271 A000272

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms, new description and formula from Christian G. Bower, Nov 15 1999

Status

approved