A317355 - OEIS

%I #14 Aug 10 2018 12:06:28

%S 1,1,5,85,5261,549061,79707245,15531175045,3926159465261,

%T 1249497583485061,488841071584907885,230674363972514998405,

%U 129251110556658394610861,84870052450743141454787461,64574784437643167984687238125,56377769340759003121860283852165,55996026841326090728124344073814061

%N E.g.f. satisfies: A(x) = Sum_{n>=0} ( exp(n*x) - A(x) )^n  / (2 - exp(n*x)*A(x))^(n+1).

%C E.g.f. A(x) = G(exp(x) - 1), where G(x) is the g.f. of A317350.

%H Vaclav Kotesovec, <a href="/A317355/b317355.txt">Table of n, a(n) for n = 0..144</a>

%F E.g.f. A(x) satisfies:

%F (1) A(x) = Sum_{n>=0} ( exp(n*x) - A(x) )^n  / (2 - exp(n*x)*A(x))^(n+1),

%F (2) A(x) = Sum_{n>=0} ( exp(n*x) + A(x) )^n  / (2 + exp(n*x)*A(x))^(n+1).

%F a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317904 = 3.9561842030261697545408... and c = 0.16545672527... - _Vaclav Kotesovec_, Aug 10 2018

%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 85*x^3/3! + 5261*x^4/4! + 549061*x^5/5! + 79707245*x^6/6! + 15531175045*x^7/7! + 3926159465261*x^8/8! + 1249497583485061*x^9/9! + ...

%e such that A = A(x) satisfies

%e A(x) = 1/(2 - A)  +  (exp(x) - A)/(2 - exp(x)*A)^2  +  (exp(2*x) - A)^2/(2 - exp(2*x)*A)^3  +  (exp(3*x) - A)^3/(2 - exp(3*x)*A)^4  +  (exp(4*x) - A)^4/(2 - exp(4*x)*A)^5  +  (exp(5*x) - A)^5/(2 - exp(5*x)*A)^6 + ...

%e Also,

%e A(x) = 1/(2 + A)  +  (exp(x) + A)/(2 + exp(x)*A)^2  +  (exp(2*x) + A)^2/(2 + exp(2*x)*A)^3  +  (exp(3*x) + A)^3/(2 + exp(3*x)*A)^4  +  (exp(4*x) + A)^4/(2 + exp(4*x)*A)^5  +  (exp(5*x) + A)^5/(2 + exp(5*x)*A)^6 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( exp(m*x +x*O(x^#A)) - Ser(A) )^m  / (2 - exp(m*x +x*O(x^#A))*Ser(A))^(m+1) ) ) ); n!*A[n+1]}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A317356, A317350.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 02 2018