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A004113
Number of rooted trees with n nodes and 2-colored non-leaf nodes.
(Formerly M1629)
5
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
1,2
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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
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
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 *)
CROSSREFS
KEYWORD
nonn,eigen
EXTENSIONS
Extended with better description from Christian G. Bower, Apr 15 1998
STATUS
approved