A004113 Number of rooted trees with n nodes and 2-colored non-leaf nodes.

1, 2, 6, 18, 60, 204, 734, 2694, 10162, 38982, 151920, 599244, 2389028, 9608668, 38945230, 158904230, 652178206, 2690598570, 11151718166, 46412717826, 193891596436, 812748036380, 3417407089470, 14410094628558, 60920843101858, 258169745573158, 1096494947168142

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• eigen

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 = 1..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.
• N. J. A. Sloane, Transforms
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Crossrefs

Cf. A004114, A052316, A052317.

Programs

Maple

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, (add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n))/n) end end: b:= etr(a): a:= n-> `if`(n<=1, n, 2*b(n-1)): seq(a(n), n=1..30); # Alois P. Heinz, Sep 06 2008

Mathematica

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]; b = etr[a]; a[n_] := If[n <= 1, n, 2*b[n - 1]]; Table[a[n], {n, 1, 27}] (* Jean-François Alcover, Jan 29 2013, translated from Alois P. Heinz's Maple program *)

Formula

Shifts left and halves under EULER transform.

a(n) ~ c * d^n / n^(3/2), where d = 4.49415643203339504537343052838796824... and c = 0.368722987377516657464802259... - Vaclav Kotesovec, Feb 28 2014

Extensions

Extended with better description from Christian G. Bower, Apr 15 1998