A306092 - OEIS

%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