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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.

%I A213507 #23 May 08 2025 10:52:28
%S A213507 1,1,3,17,149,1809,28399,550297,12732873,343231361,10576764251,
%T A213507 367054970721,14173669352413,602974492511377,28027436035348359,
%U A213507 1413479599558432169,76879014760731439889,4486205132570631391617,279595430611791210216883,18536284947404377562405041
%N 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.
%C A213507 Compare to: exp( Sum_{n>=1} (n+1)/2*A000108(n)*x^n/n ) = Sum_{n>=0} A000108(n)*x^n.
%C A213507 Conjecture: a(n) is not divisible by {2,5,13,29,31,37,41,43,53,71,97}.
%H A213507 Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, <a href="https://mathexp.eu/bostan/diagonalsFPSAC25.pdf">Diagonals of permutahedra and associahedra</a>, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See p. 7.
%F A213507 E.g.f.: (4*exp(1-2/(sqrt(1-4*x)+1)))/(sqrt(1-4*x)+1)^2. - _Vladimir Kruchinin_, Mar 22 2013
%F A213507 a(n) ~ 2^(2*n+5/2)*n^(n-2)/exp(n+1). - _Vaclav Kotesovec_, Sep 24 2013
%F A213507 From _Peter Bala_, Nov 21 2016: (Start)
%F A213507 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.
%F A213507 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)
%e A213507 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 149*x^4/4! + 1809*x^5/5! + ...
%e A213507 such that
%e A213507 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 + ...
%t A213507 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 *)
%o A213507 (PARI) {a(n)=n!*polcoeff(exp(sum(m=1,n,binomial(2*m,m)/(m+1)*x^m/m)+x*O(x^n)),n)}
%o A213507 for(n=0,30,print1(a(n),", "))
%Y A213507 Cf. A000108, A000169, A243953.
%K A213507 nonn,easy
%O A213507 0,3
%A A213507 _Paul D. Hanna_, Jun 12 2012