A317356 E.g.f. satisfies: A(x) = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n / (2 - exp(n*x)*A(x))^(n+1).

1, 2, 14, 134, 4358, 589622, 102434534, 21285122294, 5530748479718, 1792785367579382, 711595226383338854, 339665400624638782454, 192071493764203628322278, 127053485326157331378577142, 97253813187878484942034153574, 85330814329687863076988482842614, 85104598195236153766017309663096038

Offset: 0

Author

Paul D. Hanna, Aug 02 2018

Keywords

• nonn

Comments

E.g.f. A(x) = G(exp(x) - 1), where G(x) is the g.f. of A317351.

Examples

			E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 134*x^3/3! + 4358*x^4/4! + 589622*x^5/5! + 102434534*x^6/6! + 21285122294*x^7/7! + 5530748479718*x^8/8! + 1792785367579382*x^9/9! + ...
such that A = A(x) satisfies
A(x) = 1/(2 - A)  +  (exp(2*x) - A)/(2 - exp(x)*A)^2  +  (exp(3*x) - A)^2/(2 - exp(2*x)*A)^3  +  (exp(4*x) - A)^3/(2 - exp(3*x)*A)^4  +  (exp(5*x) - A)^4/(2 - exp(4*x)*A)^5  +  (exp(6*x) - A)^5/(2 - exp(5*x)*A)^6 + ...
Also,
A(x) = 1/(2 + A)  +  (exp(2*x) + A)/(2 + exp(x)*A)^2  +  (exp(3*x) + A)^2/(2 + exp(2*x)*A)^3  +  (exp(4*x) + A)^3/(2 + exp(3*x)*A)^4  +  (exp(5*x) + A)^4/(2 + exp(4*x)*A)^5  +  (exp(6*x) + A)^5/(2 + exp(5*x)*A)^6 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..136

Crossrefs

Cf. A317356, A317351.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( exp((m+1)*x +x*O(x^#A)) - Ser(A) )^m  / (2 - exp(m*x +x*O(x^#A))*Ser(A))^(m+1) ) ) ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

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

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

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

a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317904 = 3.9561842030261697545408... and c = 0.2625457134... - Vaclav Kotesovec, Aug 10 2018