A213507 E.g.f.: exp( Sum_{n>=1} A000108(n)*x^n/n ), where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan numbers.

1, 1, 3, 17, 149, 1809, 28399, 550297, 12732873, 343231361, 10576764251, 367054970721, 14173669352413, 602974492511377, 28027436035348359, 1413479599558432169, 76879014760731439889, 4486205132570631391617, 279595430611791210216883, 18536284947404377562405041

Offset: 0

Author

Paul D. Hanna, Jun 12 2012

Keywords

• nonn
• easy

Comments

Compare to: exp( Sum_{n>=1} (n+1)/2*A000108(n)*x^n/n ) = Sum_{n>=0} A000108(n)*x^n.

Conjecture: a(n) is not divisible by {2,5,13,29,31,37,41,43,53,71,97}.

Examples

			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 + ...
		

Links

• Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, Diagonals of permutahedra and associahedra, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See p. 7.

Crossrefs

Cf. A000108, A000169, A243953.

Programs

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 *)

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),", "))

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)