OFFSET
0,3
COMMENTS
G.f. A(x) = G(log(1+x)), where G(x) is the e.g.f. of A317355.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} ( (1+x)^n - A(x) )^n / (2 - (1+x)^n*A(x))^(n+1),
(2) A(x) = Sum_{n>=0} ( (1+x)^n + A(x) )^n / (2 + (1+x)^n*A(x))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317904 = 3.956184203026... and c = 0.14581304299... - Vaclav Kotesovec, Aug 07 2018
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 200*x^4 + 4160*x^5 + 99862*x^6 + 2767792*x^7 + 87200166*x^8 + 3076185774*x^9 + 120118928740*x^10 + ...
such that A = A(x) satisfies
A(x) = 1/(2 - A) + ((1+x) - A)/(2 - (1+x)*A)^2 + ((1+x)^2 - A)^2/(2 - (1+x)^2*A)^3 + ((1+x)^3 - A)^3/(2 - (1+x)^3*A)^4 + ((1+x)^4 - A)^4/(2 - (1+x)^4*A)^5 + ((1+x)^5 - A)^5/(2 - (1+x)^5*A)^6 + ...
Also,
A(x) = 1/(2 + A) + ((1+x) + A)/(2 + (1+x)*A)^2 + ((1+x)^2 + A)^2/(2 + (1+x)^2*A)^3 + ((1+x)^3 + A)^3/(2 + (1+x)^3*A)^4 + ((1+x)^4 + A)^4/(2 + (1+x)^4*A)^5 + ((1+x)^5 + A)^5/(2 + (1+x)^5*A)^6 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( (1+x)^m - Ser(A) )^m / (2 - (1+x)^m*Ser(A))^(m+1) ) ) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 02 2018
STATUS
approved