OFFSET
0,2
COMMENTS
Compare to identity: 0 = Sum_{n=-oo..+oo} x^(2*n) * (y - x^n)^n * (y + x^n)^n, which holds 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:
(1) 2 = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^n)^n * (1 - x^n*A(x))^n.
(2) 2 = Sum_{n=-oo..+oo} x^(2*n*(n-1)) / ( (A(x) - x^n)^n * (1 - x^n*A(x))^n ).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 36*x^4 + 118*x^5 + 452*x^6 + 1652*x^7 + 6534*x^8 + 25642*x^9 + 103712*x^10 + 422734*x^11 + ...
where A = A(x) satisfies
2 = ... + x^12/((A - x^3)^3*(1 - x^3*A)^3) + x^4/((A - x^2)^2*(1 - x^2*A)^2) + 1/((A - x)*(1 - x*A)) + 1 + x^2*(A - x)*(1 - x*A) + x^4*(A - x^2)^2*(1 - x^2*A)^2 + x^6*(A - x^3)^3*(1 - x^3*A)^3 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff(2 - sum(m=-#A, #A, x^(2*m) * (Ser(A) - x^m)^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, Aug 21 2022
STATUS
approved