OFFSET
0,2
COMMENTS
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(2*n+1))^n, which holds formally for all y.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(2*n+1))^(n-1).
(2) x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n^2) / (1 - x^(2*n-1)*A(x))^(n+1).
(3) A(x) = Sum_{n=-oo..+oo} x^(3*n+1)* (A(x) - x^(2*n+1))^(n-1).
(4) A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n*(n-1)) / (1 - x^(2*n-1)*A(x))^(n+1).
(5) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n*(n-1)) / (1 - x^(2*n-1)*A(x))^n.
EXAMPLE
G.f.: A(x) = 1 + 3*x + 7*x^2 + 33*x^3 + 163*x^4 + 858*x^5 + 4708*x^6 + 26662*x^7 + 154699*x^8 + 914885*x^9 + 5494719*x^10 + ...
where A = A(x) satisfies the doubly infinite sum
1 = ... + x^(-2)*(A - x^(-3))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x)^(-1) + x*(A - x^3)^0 + x^2*(A - x^5) + x^3*(A - x^7)^2 + x^4*(A - x^9)^3 + ... + x^n * (A - x^(2*n+1))^(n-1) + ...
also,
A(x) = ... + x^24/(1 - x^(-7)*A)^(-2) - x^12/(1 - x^(-5)*A)^(-1) + x^4 - 1/(1 - x^(-1)*A) + 1/(1 - x*A)^2 - x^4/(1 - x^3*A)^3 + x^12/(1 - x^5*A)^4 - x^24/(1 - x^7*A)^5 + ... + (-1)^(n+1)*x^(2*n*(n-1))/(1 - x^(2*n-1)*A(x))^(n+1) + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, x^n * (Ser(A) - x^(2*n+1))^(n-1) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 07 2022
STATUS
approved