login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A316363 O.g.f. A(x) satisfies: Sum_{n>=1} (x + (-1)^n*A(x))^n / n  =  0. 2
1, 2, 4, 14, 52, 204, 840, 3574, 15588, 69332, 313272, 1433964, 6635400, 30988312, 145871248, 691403686, 3296979524, 15805913476, 76135613784, 368304184900, 1788518253080, 8715477003688, 42605364060656, 208878870197436, 1026781984000680, 5059692979338824, 24989145569112880, 123676728224877464, 613295203581498768, 3046761116509464624 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..200

FORMULA

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

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

(1b) 0 = arctanh(x - A(x)) - log(1 - (x + A(x))^2)/2,

(1c) 1 - (x + A(x))^2 = (1+x - A(x))/(1-x + A(x)).

(2a) A(x) = x + (x + A(x))^2/(2 - (x + A(x))^2).

(2b) 0 = (2*x + x^2 - x^3) - (2 - 2*x + x^2)*A(x) + (1+x)*A(x)^2 + A(x)^3.

(3) A(-A(-x)) = x.

(4a) A(x) = -x + 2 * Series_Reversion( x - x^2/(1 - 2*x^2) ).

(4b) A(x) = x + 2 * Series_Reversion( x/sqrt(1 + 2*x^2) - x^2 )^2.

a(n) ~ sqrt(r*(r*(1-r) + s*(1+s)) / (1 + r + 3*s)) / (sqrt(Pi) * n^(3/2) * r^n), where r = 0.1912388335306640951515262439910852999016888421453... and s = 0.444963791747610196027930141875385171928290741217... are real roots of the system of equations r^2*(-1 + r + s) = s*(-2 + s + s^2) + r*(2 + 2*s + s^2), 2*r*(1 + s) + s*(2 + 3*s) = 2 + r^2. - Vaclav Kotesovec, Jul 06 2018

EXAMPLE

G.f.: A(x) = x + 2*x^2 + 4*x^3 + 14*x^4 + 52*x^5 + 204*x^6 + 840*x^7 + 3574*x^8 + 15588*x^9 + 69332*x^10 + 313272*x^11 + 1433964*x^12 + ...

such that

0 = (x - A(x)) + (x + A(x))^2/2 + (x - A(x))^3/3 + (x + A(x))^4/4 + (x - A(x))^5/5 + (x + A(x))^6/6 + (x - A(x))^7/7 + (x + A(x))^8/8 + ...

thus arctanh(x - A(x)) - log(1 - (x + A(x))^2)/2 = 0

so that (1+x - A(x))/(1-x + A(x)) = 1 - (x + A(x))^2.

RELATED SERIES.

Series_Reversion(A(x)) = x - 2*x^2 + 4*x^3 - 14*x^4 + 52*x^5 - 204*x^6 + 840*x^7 - 3574*x^8 + ... + (-1)^(n-1)*a(n)*x^n + ...

Series_Reversion(x + A(x)) = 1/2*x - 1/4*x^2 - 1/8*x^4 - 1/16*x^6 - 1/32*x^8 - 1/64*x^10 - 1/128*x^12 - 1/256*x^14 + ... = x*(1 - x)*(2 + x)/(4 - 2*x^2); equivalently, Series_Reversion((x + A(x))/2) = x - x^2/(1 - 2*x^2).

Let F(x) be the g.f. of A317800, then A(x) = F(F(x)), where

F(x) = x + x^2 + x^3 + 4*x^4 + 10*x^5 + 33*x^6 + 105*x^7 + 354*x^8 + 1214*x^9 + 4206*x^10 + ... + A317800(n)*x^n + ...

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff(sum(m=1, #A, (x + (-1)^m*x*Ser(A))^m/m), #A)); A[n]}

for(n=1, 30, print1(a(n), ", "))

CROSSREFS

Cf. A317800.

Sequence in context: A046650 A055727 A003500 * A295760 A129876 A038055

Adjacent sequences:  A316360 A316361 A316362 * A316364 A316365 A316366

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 03 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 21 01:18 EDT 2019. Contains 321356 sequences. (Running on oeis4.)