login
A349562
Number of labeled rooted forests with 2-colored leaves.
13
1, 2, 8, 56, 576, 7872, 134656, 2771456, 66744320, 1842237440, 57354338304, 1988721131520, 76015173369856, 3175757373243392, 143980934947930112, 7040807787705663488, 369414622819764928512, 20700889684976244621312, 1233951687316746828513280, 77963762014950356953333760
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
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).
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)
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