A349562 Number of labeled rooted forests with 2-colored leaves.

1, 2, 8, 56, 576, 7872, 134656, 2771456, 66744320, 1842237440, 57354338304, 1988721131520, 76015173369856, 3175757373243392, 143980934947930112, 7040807787705663488, 369414622819764928512, 20700889684976244621312, 1233951687316746828513280, 77963762014950356953333760

Offset: 0

Author

Alexander Burstein, Nov 22 2021

Keywords

• nonn

Comments

a(n) is the number of labeled trees on vertices 0,1,...,n rooted at 0, where all leaves have 2 colors (except the singleton tree 0 has only 1 color).

Examples

			a(2)=8 counts trees 0-1-2B, 0-1-2R, 0-2-1B, 0-2-1R, 1B-0-2B, 1B-0-2R, 1R-0-2B, 1R-0-2R (where B and R stand for colors Blue and Red).
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..372
• Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A000007, A007889, A216857, A038049, A201595,

Programs

Mathematica

CoefficientList[u/.AsymptoticSolve[u-E^(x(1+u))==0,u->1,{x,0,24}][[1]],x]Factorial/@Range[0,24]
nmax = 20; CoefficientList[Series[-LambertW[-x*Exp[x]]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 25 2021 *)

PARI

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k+1)^(k-1)*x^k/(1-(k+1)*x)^(k+1))) \\ Seiichi Manyama, Nov 26 2021

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*(k+1)^(n-1).

a(n) = A216857(n+1)/(n+1).

a(n) = A038049(n+1)/(n+1) for n>=1, and a(0) = A038049(1)/2.

a(n) = 2*A201595(n) - A000007(n).

E.g.f. satisfies: A(x) = e^(x*(1 + A(x))).

E.g.f. satisfies: A(-x*A(x)) = 1/A(x).

From Vaclav Kotesovec, Nov 25 2021: (Start)

E.g.f.: -LambertW(-x*exp(x))/x.

a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n) * LambertW(exp(-1))^(n+1)).

(End)

From Seiichi Manyama, Nov 26 2021: (Start)

G.f.: Sum_{k>=0} (k+1)^(k-1) * x^k/(1 - (k+1)*x)^(k+1).

a(n) = 2^n * A007889(n). (End)