OFFSET
0,3
COMMENTS
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds for all y as a formal power series in x.
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
(1) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) - (-x)^n)^(n+2) / A(x)^n.
(2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)) * A(x)^n / (1 - (-x)^n*A(x))^(n-2).
a(n) ~ c * d^n / n^(3/2), where d = 2.71312501383... and c = 3.43853109... - Vaclav Kotesovec, Mar 31 2023
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 27*x^5 + 57*x^6 + 123*x^7 + 280*x^8 + 666*x^9 + 1614*x^10 + 3955*x^11 + 9733*x^12 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = -polcoeff( sum(m=-#A, #A, (-x)^m * (1 - (-x)^m/Ser(A))^(m+2) ), #A-3)); A[n+1]}
for(n=0, 35, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 26 2023
STATUS
approved