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%I #4 Jun 22 2018 12:40:28
%S -1,3,-30,840,-45360,3963960,-512431920,91708016400,-21708518832000,
%T 6566197230552960,-2470377569057798400,1131411784221938419200,
%U -619741850665486348800000,400063411654998957081216000,-300571110264723992167009536000,260020540519396684696076728320000,-256606704941070116606793272893440000,286541492507208304817420296882114560000
%N G.f. A(x) satisfies: (1 + A(x))^A(x) = (1+x)^x, where A(x) = Sum_{n>=1} a(n)*x^n/(2*n-1)!.
%C The g.f. is also given by: A(x) = Sum_{n>=0} A306090(n)/A306091(n) * x^n.
%H Paul D. Hanna, <a href="/A306092/b306092.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) = Sum_{n>=1} a(n)*x^n/(2*n-1)! satisfies:
%F (1) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k)*x + k*A(x) = 1.
%F (2) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k)*x + (k - p)*A(x) = (1 + x)^p.
%F (3) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k - m)*x + k*A(x) = (1 + A(x))^m.
%F (4) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k - m)*x + (k - p)*A(x) = (1+x)^p * (1 + A(x))^m.
%F (5) A(A(x)) = x.
%F (6) (1 + A(x))^A(x) = (1 + x)^x.
%e G.f.: A(x) = -x + 3*x^2/3! - 30*x^3/5! + 840*x^4/7! - 45360*x^5/9! + 3963960*x^6/11! - 512431920*x^7/13! + 91708016400*x^8/15! - 21708518832000*x^9/17! + 6566197230552960*x^10/19! - 2470377569057798400*x^11/21! + 1131411784221938419200*x^12/23! - 619741850665486348800000*x^13/25! + ...
%e such that
%e (E.1) 1 = 1 + (x + A(x)) + (x + 2*A(x))*(2*x + A(x))/2! + (x + 3*A(x))*(2*x + 2*A(x))*(3*x + A(x))/3! + (x + 4*A(x))*(2*x + 3*A(x))*(3*x + 2*A(x))*(4*x + A(x))/4! + (x + 5*A(x))*(2*x + 4*A(x))*(3*x + 3*A(x))*(4*x + 2*A(x))*(5*x + A(x))/5! + ...
%e (E.2) (1 + x)^p = 1 + (x + (1-p)*A(x)) + (x + (2-p)*A(x))*(2*x + (1-p)*A(x))/2! + (x + (3-p)*A(x))*(2*x + (2-p)*A(x))*(3*x + (1-p)*A(x))/3! + (x + (4-p)*A(x))*(2*x + (3-p)*A(x))*(3*x + (2-p)*A(x))*(4*x + (1-p)*A(x))/4! + ...
%e (E.3) (1 + A(x))^m = 1 + ((1-m)*x + A(x)) + ((1-m)*x + 2*A(x))*((2-m)*x + A(x))/2! + ((1-m)*x + 3*A(x))*((2-m)*x + 2*A(x))*((3-m)*x + A(x))/3! + ((1-m)*x + 4*A(x))*((2-m)*x + 3*A(x))*((3-m)*x + 2*A(x))*((4-m)*x + A(x))/4! + ...
%e FUNCTIONAL EQUATION.
%e The series A(x) satisfies:
%e (E.4) (1 + A(x))^A(x) = (1 + x)^x = 1 + x^2 - 1/2*x^3 + 5/6*x^4 - 3/4*x^5 + 33/40*x^6 - 5/6*x^7 + 2159/2520*x^8 - 209/240*x^9 + ...
%e GENERATING METHOD.
%e Although the functional equation (1 + A(x))^A(x) = (1 + x)^x has an infinite number of solutions, one may arrive at the g.f. A(x) by the following iteration.
%e If we start with A = -x, and iterate
%e (E.5) A = (A + x*log(1 + x)/log(1 + A))/2
%e then A will converge to g.f. A(x).
%o (PARI) /* From Functional Equation (1 + A(x))^A(x) = (1 + x)^x */
%o {a(n) = my(A = -x +x*O(x^n)); for(i=1, n, A = (A + x*log(1+x +x*O(x^n))/log(1+A))/2 ); (2*n-1)! * polcoeff(A, n)}
%o for(n=1, 20, print1(a(n), ", "))
%Y Cf. A306090, A306091.
%K sign
%O 1,2
%A _Paul D. Hanna_, Jun 22 2018
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