OFFSET
0,3
COMMENTS
a(n) is the total number of leaves in all labeled forests with n nodes. Cf. A055541. - Geoffrey Critzer, Aug 22 2012
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..150
FORMULA
E.g.f.: exp(-x*LambertW(-x)).
E.g.f.: ( LambertW(-x)/(-x) )^x.
E.g.f.: ( Sum_{n>=0} (n + 1)^(n-1) * x^n/n! )^x.
E.g.f.: ( Sum_{n>=0} (n + x)^n * x^n/n! ) * (-x)/LambertW(-x). - Paul D. Hanna, Jun 16 2018
E.g.f.: LambertW(-x) / ( -x * Sum_{n>=0} (n - x)^n * x^n/n! ). - Paul D. Hanna, Jun 16 2018
a(n) = Sum_{k=0..floor(n/2)} C(n,k)*C(n-k-1,k-1)*(n-k)^(n-2*k)*k!. - Alois P. Heinz, Aug 22 2012
a(n) ~ exp(exp(-1)-1)*n^(n-1). - Vaclav Kotesovec, Jun 26 2013
EXAMPLE
E.g.f.: A(x) = 1 + 2*x^2/2! + 6*x^3/3! + 48*x^4/4! + 440*x^5/5! + ...
where
A(x) = 1 + x*(1+x)^0*x^1/1! + x*(2+x)*x^2/2! + x*(3+x)^2*x^3/3! + x*(4+x)^3*x^4/4! + ...
Also, A(x) = W(x)^x where W(x) = LambertW(-x)/(-x) and begins:
W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! + ...
MAPLE
a:= n-> add(binomial(n, k)*binomial(n-k-1, k-1)*(n-k)^(n-2*k) *k!, k=0..n/2):
seq(a(n), n=0..30); # Alois P. Heinz, Aug 22 2012
MATHEMATICA
nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
Range[0, nn]! CoefficientList[Series[Exp[x t] , {x, 0, nn}], x] (* Geoffrey Critzer, Aug 22 2012 *)
PROG
(PARI) {a(n)=local(A=sum(k=0, n, x*(k+x)^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(A, n)}
(PARI) {a(n)=local(W=sum(k=0, n, (k+1)^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(W^x, n)}
(PARI) {a(n)=local(W=sum(k=0, n, (k+1)^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(exp(x^2*W), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 13 2011
STATUS
approved