OFFSET
0,3
COMMENTS
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
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 the following.
(1) x = Sum_{n=-oo..+oo} x^n * (1 - x^n * A(-x)^n)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * 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 = 5.390297559554269719991046... and c = 0.267652299887938085649... - Vaclav Kotesovec, Dec 25 2022
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 21*x^4 + 88*x^5 + 377*x^6 + 1654*x^7 + 7424*x^8 + 34000*x^9 + 158274*x^10 + 746525*x^11 + 3559456*x^12 + ...
SPECIFIC VALUES.
A(x) = 3/2 at x = 0.1850570503493984408934312903280642188437354418734...
A(1/6) = 1.3085832721715442420948608003299892250459754159045...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(x + sum(n=-#A, #A, (-x)^n * (1 - (-x)^n * Ser(A)^n )^n ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 24 2022
STATUS
approved