OFFSET
0,3
COMMENTS
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n)^n, which holds 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 may be defined by the following.
(1) x = Sum_{n=-oo..+oo} x^n * (A(-x) - x^n)^n / A(-x)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(-x)^n / (1 - x^n*A(-x))^n.
a(n) ~ c * d^n / n^(3/2), where d = 4.00524760257508238375... and c = 0.76876562144270017... - Vaclav Kotesovec, Feb 13 2023
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 49*x^5 + 159*x^6 + 528*x^7 + 1784*x^8 + 6145*x^9 + 21439*x^10 + 75654*x^11 + 269525*x^12 + ...
such that x = P(x) + Q(x) where
P(x) = x*(1 - x/A(-x)) + x^2*(1 - x^2/A(-x))^2 + x^3*(1 - x^3/A(-x))^3 + x^4*(1 - x^4/A(-x))^4 + ... + x^n * (1 - x^n/A(-x))^n + ...
Q(x) = 1 - A(-x)/(1 - x*A(-x)) + x^2*A(-x)^2/(1 - x^2*A(-x))^2 - x^6*A(-x)^3/(1 - x^3*A(-x))^3 + ... + (-1)^n*x^(n*(n-1))*A(-x)^n/(1 - x^n*A(-x))^n + ...
PROG
(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff(-x - sum(m=-#A, #A, (-x)^m*(1 - (-x)^m/Ser(A))^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff(-x - sum(m=-#A, #A, (-1)^m * x^(m*(m-1))*Ser(A)^m/(1 - (-x)^m*Ser(A))^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 13 2023
STATUS
approved