OFFSET
0,2
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).
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.
FORMULA
a(n) = Sum_{k=0..n} binomial(n,k)*(k+1)^(n-1).
a(n) = A216857(n+1)/(n+1).
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)
EXAMPLE
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).
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Alexander Burstein, Nov 22 2021
STATUS
approved