The On-Line Encyclopedia of Integer Sequences (OEIS)

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A304639

G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n.
(history; published version)

#23 by Paul D. Hanna at Mon Dec 27 20:42:00 EST 2021

STATUS

editing

approved

#22 by Paul D. Hanna at Mon Dec 27 20:41:35 EST 2021

FORMULA

G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 117*x^4 + 1735*x^5 + 31853*x^6 + 689043*x^7 + 17079221*x^8 + 476238926*x^9 + 14742680162*x^10 + 501584454703*x^11 + ...

is such that

1 = 1 + (1/(1-x) - A(x)) + (1/(1-x)^2 - A(x))^2 + (1/(1-x)^3 - A(x))^3 + (1/(1-x)^4 - A(x))^4 + (1/(1-x)^5 - A(x))^5 + (1/(1-x)^6 - A(x))^6 + (1/(1-x)^7 - A(x))^7 + ...

Also,

1 = 1/(1 + A(x)) + (1-x)/((1-x) + A(x))^2 + (1-x)^2/((1-x)^2 + A(x))^3 + (1-x)^3/((1-x)^3 + A(x))^4 + (1-x)^4/((1-x)^4 + A(x))^5 + (1-x)^5/((1-x)^5 + A(x))^6 + (1-x)^6/((1-x)^6 + A(x))^7 + ...

PARTICULAR VALUES.

Although the power series A(x) diverges at x = -1, it may be evaluated formally.

Let t = A(-1) = 0.5452189736359494312349502450349441069576127988881794567242641...

then t satisfies

(1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.

(2) 1 = Sum_{n>=0} ( 1 - 2^n*t )^n / 2^(n^2).

(3) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 117*x^4 + 1735*x^5 + 31853*x^6 + 689043*x^7 + 17079221*x^8 + 476238926*x^9 + 14742680162*x^10 + 501584454703*x^11 + ...

is such that

1 = 1 + (1/(1-x) - A(x)) + (1/(1-x)^2 - A(x))^2 + (1/(1-x)^3 - A(x))^3 + (1/(1-x)^4 - A(x))^4 + (1/(1-x)^5 - A(x))^5 + (1/(1-x)^6 - A(x))^6 + (1/(1-x)^7 - A(x))^7 + ...

Also,

1 = 1/(1 + A(x)) + (1-x)/((1-x) + A(x))^2 + (1-x)^2/((1-x)^2 + A(x))^3 + (1-x)^3/((1-x)^3 + A(x))^4 + (1-x)^4/((1-x)^4 + A(x))^5 + (1-x)^5/((1-x)^5 + A(x))^6 + (1-x)^6/((1-x)^6 + A(x))^7 + ...

PARTICULAR VALUES.

Although the power series A(x) diverges at x = -1, it may be evaluated formally.

Let t = A(-1) = 0.5452189736359494312349502450349441069576127988881794567242641...

then t satisfies

(1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.

(2) 1 = Sum_{n>=0} ( 1 - 2^n*t )^n / 2^(n^2).

(3) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).

STATUS

approved

editing

Discussion

Mon Dec 27

20:42

Paul D. Hanna: Moved misplaced examples to the example section.

#21 by N. J. A. Sloane at Fri Dec 17 01:03:49 EST 2021

STATUS

editing

approved

#20 by N. J. A. Sloane at Fri Dec 17 01:03:45 EST 2021

FORMULA

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n.

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

(3) 1 = Sum_{n>=0} (1-x)^n / ( (1-x)^n + A(x) )^(n+1).

G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 117*x^4 + 1735*x^5 + 31853*x^6 + 689043*x^7 + 17079221*x^8 + 476238926*x^9 + 14742680162*x^10 + 501584454703*x^11 + ...

is such that

1 = 1 + (1/(1-x) - A(x)) + (1/(1-x)^2 - A(x))^2 + (1/(1-x)^3 - A(x))^3 + (1/(1-x)^4 - A(x))^4 + (1/(1-x)^5 - A(x))^5 + (1/(1-x)^6 - A(x))^6 + (1/(1-x)^7 - A(x))^7 + ...

Also,

1 = 1/(1 + A(x)) + (1-x)/((1-x) + A(x))^2 + (1-x)^2/((1-x)^2 + A(x))^3 + (1-x)^3/((1-x)^3 + A(x))^4 + (1-x)^4/((1-x)^4 + A(x))^5 + (1-x)^5/((1-x)^5 + A(x))^6 + (1-x)^6/((1-x)^6 + A(x))^7 + ...

PARTICULAR VALUES.

Although the power series A(x) diverges at x = -1, it may be evaluated formally.

Let t = A(-1) = 0.5452189736359494312349502450349441069576127988881794567242641...

then t satisfies

(1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.

(2) 1 = Sum_{n>=0} ( 1 - 2^n*t )^n / 2^(n^2).

(3) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).

EXAMPLE

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n.

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

(3) 1 = Sum_{n>=0} (1-x)^n / ( (1-x)^n + A(x) )^(n+1).

MAPLE

G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 117*x^4 + 1735*x^5 + 31853*x^6 + 689043*x^7 + 17079221*x^8 + 476238926*x^9 + 14742680162*x^10 + 501584454703*x^11 + ...

such that

1 = 1 + (1/(1-x) - A(x)) + (1/(1-x)^2 - A(x))^2 + (1/(1-x)^3 - A(x))^3 + (1/(1-x)^4 - A(x))^4 + (1/(1-x)^5 - A(x))^5 + (1/(1-x)^6 - A(x))^6 + (1/(1-x)^7 - A(x))^7 + ...

Also,

1 = 1/(1 + A(x)) + (1-x)/((1-x) + A(x))^2 + (1-x)^2/((1-x)^2 + A(x))^3 + (1-x)^3/((1-x)^3 + A(x))^4 + (1-x)^4/((1-x)^4 + A(x))^5 + (1-x)^5/((1-x)^5 + A(x))^6 + (1-x)^6/((1-x)^6 + A(x))^7 + ...

PARTICULAR VALUES.

Although the power series A(x) diverges at x = -1, it may be evaluated formally.

Let t = A(-1) = 0.5452189736359494312349502450349441069576127988881794567242641...

then t satisfies

(1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.

(2) 1 = Sum_{n>=0} ( 1 - 2^n*t )^n / 2^(n^2).

(3) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).

STATUS

proposed

editing

#19 by Jon E. Schoenfield at Thu Dec 16 22:25:57 EST 2021

STATUS

editing

proposed

#18 by Jon E. Schoenfield at Thu Dec 16 22:24:49 EST 2021

MAPLE

1 = 1 + (1/(1-x) - A(x)) + (1/(1-x)^2 - A(x))^2 + (1/(1-x)^3 - A(x))^3 + (1/(1-x)^4 - A(x))^4 + (1/(1-x)^5 - A(x))^5 + (1/(1-x)^6 - A(x))^6 + (1/(1-x)^7 - A(x))^7 + ...

STATUS

approved

editing

Discussion

Thu Dec 16

22:25

Jon E. Schoenfield: I don't know Maple, but it's very hard for me to believe that the Maple section here contains real Maple code. (Maybe artificial maple, with artificial sweeteners or something...)  ?:-/

#17 by Vaclav Kotesovec at Wed Oct 14 08:03:53 EDT 2020

STATUS

editing

approved

#16 by Vaclav Kotesovec at Wed Oct 14 08:03:34 EDT 2020

FORMULA

a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.16108865386542881383... and c = 0.16107844724485... - Vaclav Kotesovec, Oct 14 2020

STATUS

approved

editing

#15 by Paul D. Hanna at Thu Jun 20 22:49:50 EDT 2019

STATUS

editing

approved

#14 by Paul D. Hanna at Thu Jun 20 22:49:48 EDT 2019

CROSSREFS

Cf. A326262, A326263, A326264, A326265.

STATUS

approved

editing