A004114 Number of trees with n nodes and 2-colored internal (non-leaf) nodes. (Formerly M1422)

1, 1, 1, 2, 5, 12, 33, 98, 305, 1002, 3424, 12016, 43230, 158516, 590621, 2230450, 8521967, 32889238, 128064009, 502590642, 1986357307, 7900377892, 31602819524, 127076645038, 513419837168, 2083414420394, 8488377206876, 34712566540014, 142443837953632

Offset

0,4

References

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 = 0..500

F. Harary, R. W. Robinson and A. J. Schwenk, Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc., Series A, 20 (1975), 483-503. Errata: Vol. A 41 (1986), p. 325.

Index entries for sequences related to trees

Formula

G.f.: 1+B(x)-x*B(x)-B(x)^2/2+B(x^2)/2 where B(x) is g.f. of A004113. - Christian G. Bower, Dec 15 1999

a(n) ~ c * d^n / n^(5/2), where d = 4.49415643203339504537343052... (same as for A004113), c = 0.31497820931312537077... . - Vaclav Kotesovec, Sep 12 2014

Mathematica

max = 28; etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n ]; b]; bb = etr[A004113]; A004113[n_] := If[n <= 1, n, 2*bb[n - 1]]; b[x_] := Sum[A004113[n] x^n, {n, 1, max}]; f[x_] := Sum[a[n] x^n, {n, 0, max}]; a[0] = a[1] = a[2] = 1; coes = CoefficientList[ Series[f[x] - (1 + b[x] - x*b[x] - b[x]^2/2 + b[x^2]/2), {x, 0, max}], x]; Table[a[n], {n, 0, max}] /. Solve[Thread[coes == 0]][[1]] (* Jean-François Alcover, Jan 29 2013, after Alois P. Heinz *)

Crossrefs

Cf. A004113, A052316, A052317.

Sequence in context: A225616 A186739 A266292 * A208957 A209051 A209216

Adjacent sequences: A004111 A004112 A004113 * A004115 A004116 A004117

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved