|
| |
|
|
A316363
|
|
O.g.f. A(x) satisfies: Sum_{n>=1} (x + (-1)^n*A(x))^n / n = 0.
|
|
4
|
|
|
|
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
|
|
|
|
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.
(5) (A'(x) - 1)/(A'(x) + 1) = (x + A(x))*(1 - (x - A(x))^2) / (1 - (x + A(x))^2). - Paul D. Hanna, Apr 28 2022
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
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
