OFFSET
0,3
COMMENTS
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(3*n+2))^n, which holds formally 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 the following formulas.
(1) 1 = Sum_{n=-oo..+oo} x^n * (x^(3*n+1) - A(x))^n.
(2) 1 = Sum_{n=-oo..+oo} x^(n*(3*n-2)) / (1 - x^(3*n-1)*A(x))^n.
(3) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (x^(3*n+2) - A(x))^n (trivial).
(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n*(n-1)) / (1 - x^(3*n-1)*A(x))^n (trivial).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 23*x^5 + 55*x^6 + 138*x^7 + 349*x^8 + 904*x^9 + 2377*x^10 + 6323*x^11 + 16993*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^(3*n+1) - Ser(A))^n ), #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 29 2023
STATUS
approved