OFFSET
0,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^n )^n.
(2) A(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(n+1) )^n.
(3) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(n+1) )^n / (1+x)^(n+1).
a(n) ~ n^n / (2^(log(2)/2 + 5/2) * sqrt(1-log(2)) * exp(n) * (log(2))^(2*n + 1)). - Vaclav Kotesovec, Aug 12 2018
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 26*x^4 + 239*x^5 + 2768*x^6 + 38267*x^7 + 611193*x^8 + 11040954*x^9 + 222241117*x^10 + ...
such that
1 = 1 + (1/A(x) - 1/(1+x)) + (1/A(x) - 1/(1+x)^2)^2 + (1/A(x) - 1/(1+x)^3)^3 + (1/A(x) - 1/(1+x)^4)^4 + (1/A(x) - 1/(1+x)^5)^5 + (1/A(x) - 1/(1+x)^6)^6 + (1/A(x) - 1/(1+x)^7)^7 + (1/A(x) - 1/(1+x)^8)^8 + ...
Also,
A(x) = 1 + (1/A(x) - 1/(1+x)^2) + (1/A(x) - 1/(1+x)^3)^2 + (1/A(x) - 1/(1+x)^4)^3 + (1/A(x) - 1/(1+x)^5)^4 + (1/A(x) - 1/(1+x)^6)^5 + (1/A(x) - 1/(1+x)^7)^6 + (1/A(x) - 1/(1+x)^8)^7 + (1/A(x) - 1/(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/(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 10 2018
STATUS
approved