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%I #16 Feb 26 2014 10:19:17
%S 1,1,3,25,349,6821,171421,5265625,191160201,8007548617,380157603481,
%T 20171371753061,1182973489103869,75984447924612397,
%U 5305029326492409333,400014338565211619761,32396515980658185762961
%N E.g.f. satisfies: A(x) = exp( x*Catalan(x*A(x)) ), where Catalan(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of A000108.
%F E.g.f.: A(x) = exp(F(x)) where F(x) = x + F(x)^2*exp(F(x)) is the e.g.f. of A213643.
%F E.g.f.: A(x) = Sum_{n>=0} a(n)*x^n/n!, where
%F a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(2*n-k,n-k)*k/(2*n-k).
%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
%F a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(2*n-k,n-k)*k/(2*n-k).
%F ...
%F O.g.f.: A(x) = 1 + Sum_{n>=1} (2*n-2)!/(n-1)! * x^n/(1 - n*x)^(2*n-1).
%F a(n) ~ n^(n-1) * sqrt((r*s^3*(1-6*r*s+8*r^2*s^2)) / (1 - (-2+8*r+r^2)*s + 4*r*(-4+4*r+r^2)*s^2 + 4*r^2*(8+r)*s^3)) / (exp(n) * r^n), where s = 1.370489293947401403417767032... is the root of the equation log(s)*(1-s*log(s)) + 2*(1+s) = (1+2*s) * sqrt((1+s)/s), and r = log(s)*(1-s*log(s)) = 0.179036084709909351719214... - _Vaclav Kotesovec_, Feb 26 2014
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 349*x^4/4! + 6821*x^5/5! +...
%e log(A(x))/x = 1 + x*A(x) + 2*x^2*A(x)^2 + 5*x^3*A(x)^3 + 14*x^4*A(x)^4 +...+ A000108(n)*x^n*A(x)^n +...
%e log(A(x))/x = 1 + x + 6*x^2/2! + 63*x^3/3! + 988*x^4/4! + 20725*x^5/5! +...+ A213644(n)*x^n/n! +...
%e log(A(x)) = x + 2*x^2/2! + 18*x^3/3! + 252*x^4/4! + 4940*x^5/5! +...+ A213643(n)*x^n/n! +...
%e Ordinary Generating Function:
%e O.g.f.: 1 + x + 3*x^2 + 25*x^3 + 349*x^4 + 6821*x^5 + 171421*x^6 +...
%e O.g.f.: 1 + x/(1-x) + 2*x^2/(1-2*x)^3 + 6*2!*x^3/(1-3*x)^5 + 20*3!*x^4/(1-4*x)^7 + 70*4!*x^5/(1-5*x)^9 + 252*5!*x^6/(1-6*x)^11 +...+ (2*n-2)!/(n-1)!*x^n/(1-n*x)^(2*n-1) +...
%t Flatten[{1,Table[Sum[n!/k!*(n-k+1)^(k-1)*Binomial[2*n-k, n-k]*k/(2*n-k),{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 26 2014 *)
%o (PARI) {a(n,m=1)=if(n==0,1,sum(k=0,n,n!/k!*m*(m+n-k)^(k-1)*binomial(2*n-k,n-k)*k/(2*n-k)))}
%o (PARI) {a(n,m=1)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp((1-sqrt(1-4*x*A))/(2*A)));n!*polcoeff(A^m,n)}
%o (PARI) /* O.g.f.: */
%o {a(n)=polcoeff(1+sum(m=1, n, (2*m-2)!/(m-1)!*x^m/(1-m*x+x*O(x^n))^(2*m-1)), n)}
%Y Cf. A213643 (log), A214689, A000108.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 17 2009
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