A306092 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)!.

-1, 3, -30, 840, -45360, 3963960, -512431920, 91708016400, -21708518832000, 6566197230552960, -2470377569057798400, 1131411784221938419200, -619741850665486348800000, 400063411654998957081216000, -300571110264723992167009536000, 260020540519396684696076728320000, -256606704941070116606793272893440000, 286541492507208304817420296882114560000

Offset

1,2

Comments

The g.f. is also given by: A(x) = Sum_{n>=0} A306090(n)/A306091(n) * x^n.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n/(2*n-1)! satisfies:

(1) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k)*x + k*A(x) = 1.

(2) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k)*x + (k - p)*A(x) = (1 + x)^p.

(3) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k - m)*x + k*A(x) = (1 + A(x))^m.

(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.

(5) A(A(x)) = x.

(6) (1 + A(x))^A(x) = (1 + x)^x.

Example

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! + ...
such that
(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.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.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! + ...
FUNCTIONAL EQUATION.
The series A(x) satisfies:
(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 + ...
GENERATING METHOD.
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.
If we start with A = -x, and iterate
(E.5) A = (A + x*log(1 + x)/log(1 + A))/2
then A will converge to g.f. A(x).

Prog

(PARI) /* From Functional Equation (1 + A(x))^A(x) = (1 + x)^x */
{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)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A306090, A306091.

Sequence in context: A065753 A255926 A113677 * A174549 A363426 A184575

Adjacent sequences: A306089 A306090 A306091 * A306093 A306094 A306095

Keyword

sign

Author

Paul D. Hanna, Jun 22 2018

Status

approved