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A029856 Number of rooted trees with 2-colored leaves. 8
2, 2, 5, 13, 37, 108, 332, 1042, 3360, 11019, 36722, 123875, 422449, 1453553, 5040816, 17599468, 61814275, 218252584, 774226549, 2758043727, 9862357697, 35387662266, 127374191687, 459783039109, 1664042970924, 6037070913558, 21951214425140, 79981665585029 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 768

N. J. A. Sloane, Transforms

Index entries for sequences related to rooted trees

FORMULA

Shifts left under Euler transform.

G.f. satisfies: A(x) = x + x*exp( Sum_{n>=1} A(x^n)/n ). - Paul D. Hanna, Oct 19 2005

a(n) ~ c * d^n / n^(3/2), where d = 3.848442876944251389076286931217197... and c = 0.48335853985605895591573724406549734... - Vaclav Kotesovec, Mar 29 2014

MAPLE

A:= proc(n) option remember; if n=0 then 0 else convert(series(x+x* exp(sum(subs(x=x^i, A(n-1))/i, i=1..n-1)), x=0, n+1), polynom) fi end; a:= n-> coeff(A(n), x, n): seq(a(n), n=1..25); # Alois P. Heinz, Aug 22 2008

# second Maple program:

with(numtheory): a:= proc(n) option remember; local d, j; if n<=1 then 2*n else (add(d*a(d), d=divisors(n-1)) +add(add(d*a(d), d=divisors(j)) *a(n-j), j=1..n-2))/ (n-1) fi end: seq(a(n), n=1..25); # Alois P. Heinz, Sep 06 2008

MATHEMATICA

a[n_] := a[n] = If [n <= 1, 2*n, (Sum[d*a[d], {d, Divisors[n-1]}] + Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j], {j, 1, n-2}])/(n-1)]; Array[a, 25] (* Jean-Fran├žois Alcover, Mar 13 2015, after Alois P. Heinz *)

PROG

(PARI) {a(n)=local(A=x+x*O(x^n)); for(i=1, n, A=x+x*exp(sum(m=1, n, subst(A, x, x^m)/m))); polcoeff(A, n, x)} (Hanna)

CROSSREFS

Essentially the same as A036249. Cf. A000081, A029857, A038049.

Sequence in context: A208175 A078413 A019083 * A072898 A032290 A032201

Adjacent sequences:  A029853 A029854 A029855 * A029857 A029858 A029859

KEYWORD

nonn,easy,eigen

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified January 19 17:45 EST 2019. Contains 319309 sequences. (Running on oeis4.)