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+x)^n - 1)^n/(A(x) + 1 - (1+x)^n)^(n+1).
(2) 1 = Sum_{n>=0} ((1+x)^n + 1)^n/(A(x) + 1 + (1+x)^n)^(n+1).
a(n) ~ c * A317904^n * n^n / exp(n), where c = 0.546536986945124389069... - Vaclav Kotesovec, Aug 11 2021
EXAMPLE
G.f.: A(x) = 1 + x + 5*x^2 + 42*x^3 + 553*x^4 + 9757*x^5 + 213989*x^6 + 5577285*x^7 + 167819725*x^8 + 5715066723*x^9 + 217100774130*x^10 + ...
such that
1 = 1/A(x) + ((1+x) - 1)/(A(x) + 1 - (1+x))^2 + ((1+x)^2 - 1)^2/(A(x) + 1 - (1+x)^2)^3 + ((1+x)^3 - 1)^3/(A(x) + 1 - (1+x)^3)^4 + ((1+x)^4 - 1)^4/(A(x) + 1 - (1+x)^4)^5 + ((1+x)^5 - 1)^5/(A(x) + 1 - (1+x)^5)^6 + ...
also,
1 = 1/(A(x) + 2) + (1 + (1+x))/(A(x) + 1 + (1+x))^2 + (1 + (1+x)^2)^2/(A(x) + 1 + (1+x)^2)^3 + (1 + (1+x)^3)^3/(A(x) + 1 + (1+x)^3)^4 + (1 + (1+x)^4)^4/(A(x) + 1 + (1+x)^4)^5 + (1 + (1+x)^5)^5/(A(x) + 1 + (1+x)^5)^6 + ...
PROG
(PARI) {a(n) = my(A=[1], X=x+x*O(x^n)); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+X)^m - 1)^m / (Ser(A) + 1 - (1+X)^m)^(m+1) ) )[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 02 2019
STATUS
approved