OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^n * (1 + x^n*A(x)^n)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^(n*(n-1)) * A(x)^(n^2) / (1 + x^n*A(x)^n)^n.
a(n) ~ c * d^n / n^(3/2), where d = 4.76347639696677679... and c = 0.37393658540119283... - Vaclav Kotesovec, Jan 11 2023
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 21*x^4 + 72*x^5 + 257*x^6 + 998*x^7 + 3988*x^8 + 16064*x^9 + 65734*x^10 + 273541*x^11 + 1151184*x^12 + ...
where
x = ... + x^6*A(x)^9/(1 + x^3*A(x)^3)^3 - x^2*A(x)^4/(1 + x^2*A(x)^2)^2 + A(x)/(1 + x*A(x)) - 1 + x*(1 + x*A(x)) - x^2*(1 + x^2*A(x)^2)^2 + x^3*(1 + x^3*A(x)^3)^3 + ... + (-1)^(n-1) * x^n * (1 + x^n*A(x)^n)^n + ...
SPECIFIC VALUES.
A(1/d) = 1.71831164... where d = 4.76347639696677679... is given in the formula section.
A(1/5) = 1.47621312973364884841150188176844829427560286588046...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(x - sum(m=-#A, #A, (-1)^(m-1) * x^m * (1 + (x*Ser(A))^m)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 10 2023
STATUS
approved