|
| |
|
|
A304639
|
|
G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n.
|
|
8
|
|
|
|
1, 1, 2, 11, 117, 1735, 31853, 689043, 17079221, 476238926, 14742680162, 501584454703, 18605089712174, 747393133162471, 32332767332220442, 1498961537925543920, 74153115616699819304, 3899494667155151052688, 217246028175467702590241, 12783023090792392539557926, 792236994094236725330142276, 51585659784100723438219893047, 3520987513029712770759434038820
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
LINKS
|
|
|
|
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).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.16108865386542881383... and c = 0.16107844724485... - Vaclav Kotesovec, Oct 14 2020
|
|
|
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).
|
|
|
PROG
|
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (1/(1-x +x^2*O(x^n))^m - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
