OFFSET
0,3
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 the following formulas.
(1) 1 = Sum_{n=-oo..+oo} x^n * (x^(2*n) - A(x))^n.
(2) 1 = Sum_{n=-oo..+oo} x^(n*(2*n-1)) / (1 - x^(2*n)*A(x))^n ).
(3) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (x^(2*n+1) - A(x))^n (trivial).
(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 - x^(2*n)*A(x))^n ) (trivial).
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 13*x^4 + 31*x^5 + 76*x^6 + 192*x^7 + 504*x^8 + 1351*x^9 + 3668*x^10 + 10082*x^11 + 27991*x^12 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, x^n * (x^(2*n) - Ser(A))^n ), #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 07 2022
STATUS
approved