OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^n )^n.
(2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(n+1) )^n.
(3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(n+1) )^n * (1-x)^(n+1).
(4) A(x)^2 = 2*A(x) * [ Sum_{n>=0} (n+1) * ( 1/A(x) - (1-x)^(n+1) )^n ] - [ Sum_{n>=0} (n+1) * ( 1/A(x) - (1-x)^(n+2) )^n ].
(5) A(x) = [ Sum_{n>=1} n*(n+1)/2 * (1-x)^(n+1) * ( 1/Ser(A) - (1-x)^(n+1) )^(n-1) ] / [ Sum_{n>=1} n^2 * (1-x)^n * ( 1/Ser(A) - (1-x)^n )^(n-1) ].
a(n) ~ 2^(log(2)/2 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 06 2018
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 42*x^4 + 372*x^5 + 4269*x^6 + 59047*x^7 + 946557*x^8 + 17175289*x^9 + 347208299*x^10 + ...
such that
1 = 1 + (1/A(x) - (1-x)) + (1/A(x) - (1-x)^2)^2 + (1/A(x) - (1-x)^3)^3 + (1/A(x) - (1-x)^4)^4 + (1/A(x) - (1-x)^5)^5 + (1/A(x) - (1-x)^6)^6 + (1/A(x) - (1-x)^7)^7 + (1/A(x) - (1-x)^8)^8 + ...
Also,
A(x) = 1 + (1/A(x) - (1-x)^2) + (1/A(x) - (1-x)^3)^2 + (1/A(x) - (1-x)^4)^3 + (1/A(x) - (1-x)^5)^4 + (1/A(x) - (1-x)^6)^5 + (1/A(x) - (1-x)^7)^6 + (1/A(x) - (1-x)^8)^7 + (1/A(x) - (1-x)^9)^8 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( 1/Ser(A) - (1-x)^(m+1) )^m ) )[#A]/2 ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 02 2018
STATUS
approved