OFFSET
0,4
COMMENTS
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n*F(x)^n)^n / (1 - x*F(x)^2)^n, where F(x) = 1 + x*F(x)^3.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
Given F(x) = 1 + x*F(x)^3, g.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = Sum_{n>=0} x^n * F(x)^n * (1 - x^n*F(x)^n)^n.
(2) A(x) = Sum_{n>=1} (-1)^(n-1) * (x*F(x))^(n*(n-1)) / (1 - x^n*F(x)^n)^n.
(3) A(x) = Sum_{n>=0} x^n * (1 - x^n*F(x)^n)^n / (1 - x*F(x)^2)^n.
(4) A(x) = Sum_{n>=1} (-1)^(n-1) * x^(n*(n-1)) * F(x)^(n^2) * (1 - x*F(x)^2)^n / (1 - x^n*F(x)^n)^n.
a(n) ~ c * 3^(3*n) / (n^(3/2) * 2^(2*n)), where c = 0.0403028056146458801802487899052088995113692232406693619.... - Vaclav Kotesovec, Mar 14 2023
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 14*x^4 + 64*x^5 + 314*x^6 + 1633*x^7 + 8826*x^8 + 49107*x^9 + 279349*x^10 + 1617290*x^11 + 9498099*x^12 + ...
where
F(x) = 1 + x*F(x)*(1 - x*F(x)) + x^2*F(x)^2*(1 - x^2*F(x)^2) + x^3*F(x)^3*(1 - x^3*F(x)^3) + x^4*F(x)^4*(1 - x^4*F(x)^4) + ... + x^n * F(x)^n * (1 - x^n*F(x)^n)^n + ...
also,
F(x) = 1/(1 - x*F(x)) - (x*F(x))^2/(1 - x^2*F(x)^2)^2 + (x*F(x))^6/(1 - x^3*F(x)^3)^3 - (x*F(x))^12/(1 - x^4*F(x)^4)^4 + (x*F(x))^20/(1 - x^5*F(x)^4)^5 +- ... + (-1)^(n-1) * (x*F(x))^(n*(n-1)) / (1 - x^n*F(x)^n)^n + ...
Where F(x) = 1 + x*F(x)^3 begins
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 + 43263*x^8 + 246675*x^9 + 1430715*x^10 + ... + A001764(n)*x^n + ...
SPECIFIC VALUES.
The radius of convergence of the power series A(x) equals 4/27.
The power series A(x) converges at x = 4/27 to
A(4/27) = 1.2311920996301390036800654138630946234233891541082821783156...
which equals the following sums:
(1) A(4/27) = Sum_{n>=0} 2^n * (9^n - 2^n)^n / 9^(n*(n+1)),
(2) A(4/27) = Sum_{n>=1} (-1)^(n-1) * 2^(n*(n-1)) * 9^n / (9^n - 2^n)^n.
PROG
(PARI) {a(n) = my(A=1, F = (serreverse(x/(1+x + O(x^(n+2)))^3)/x)^(1/3));
A = sum(m=0, n, x^m * F^m * (1 - x^m*F^m)^m); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1, F = (serreverse(x/(1+x + O(x^(n+3)))^3)/x)^(1/3));
A = sum(m=1, n+1, (-1)^(m-1) * (x*F)^(m*(m-1)) / (1 - x^m*F^m)^m); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 20 2022
STATUS
approved