OFFSET
0,2
FORMULA
G.f. A(x) satisfies:
(1) 0 = Sum_{n=-oo..+oo} x^(n^2) * (x^n - A(x))^(n+1).
(2) 0 = Sum_{n=-oo..+oo} x^(n*(2*n+1)) / (1 - A(x)*x^n)^(n-1).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 2*x^2 + 5*x^3 + 13*x^4 + 34*x^5 + 100*x^6 + 310*x^7 + 973*x^8 + 3124*x^9 + 10251*x^10 + 34086*x^11 + 114610*x^12 + ...
where
0 = ... + x^9/(1/x^3 - A(x))^2 + x^4/(1/x^2 - A(x)) + x + (1 - A(x)) + x*(x - A(x))^2 + x^4*(x^2 - A(x))^3 + x^9*(x^3 - A(x))^4 + x^16*(x^4 - A(x))^5 + ... + x^(n^2)*(x^n - A(x))^(n+1) + ...
PROG
(PARI) {a(n) = my(A=[1], M); for(i=1, n, A=concat(A, 0); M = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff( sum(m=-M, M, x^(m^2) * (x^m - Ser(A))^(m+1) ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 22 2022
STATUS
approved