OFFSET
0,3
COMMENTS
FORMULA
E.g.f.: (4*exp(1-2/(sqrt(1-4*x)+1)))/(sqrt(1-4*x)+1)^2. - Vladimir Kruchinin, Mar 22 2013
a(n) ~ 2^(2*n+5/2)*n^(n-2)/exp(n+1). - Vaclav Kotesovec, Sep 24 2013
From Peter Bala, Nov 21 2016: (Start)
Conjectural e.g.f.: 1/x * Series_Reversion ( x/(1 + F(x)) ), where F(x) = x + Sum_{n >= 2} (n - 1)^(n - 2)*x^n/n! = x + x^2/2! + 2*x^3/3! + 3^2*x^4/4! + 4^3*x^5/5! + 5^4*x^6/6! + ...; that is, dF/dx = 1 - LambertW(-x) = 1 + Euler's tree function T(x). See A000169.
The conjecture is equivalent to the result: Catalan(n) = [x^n] (1 + F(x))^n = [x^n] (2*x + x*T(x) + x/T(x))^n. (End)
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 149*x^4/4! + 1809*x^5/5! + ...
such that
log(A(x)) = x + 2*x^2/2 + 5*x^3/3 + 14*x^4/4 + 42*x^5/5 + 132*x^6/6 + 429*x^7/7 + 1430*x^8/8 + ... + A000108(n)*x^n/n + ...
MATHEMATICA
With[{nn=20}, CoefficientList[Series[(4Exp[1-2/(Sqrt[1-4x]+1)])/ (Sqrt[ 1-4x]+1)^2, {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jan 18 2016 *)
PROG
(PARI) {a(n)=n!*polcoeff(exp(sum(m=1, n, binomial(2*m, m)/(m+1)*x^m/m)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Jun 12 2012
STATUS
approved