OFFSET
0,2
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.
(1) 3 = Sum_{n=-oo..+oo} (-1)^n * x^n * (3*A(x) + x^(2*n-1))^(n+1).
(2) 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + 3*A(x)*x^(2*n+1))^(n-1).
(3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (3*A(x) + x^(2*n-1))^n.
(4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (3*A(x) + x^(2*n-1))^(n-1).
(5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + 3*A(x)*x^(2*n+1))^n.
a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) * 3^(n-2*k) for n >= 0.
EXAMPLE
G.f.: A(x) = 1 + 3*x + 11*x^2 + 45*x^3 + 193*x^4 + 846*x^5 + 3779*x^6 + 17169*x^7 + 79115*x^8 + 368820*x^9 + 1736169*x^10 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (3*Ser(A) + x^(2*m-1))^(m+1) ), #A-1)/3); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 20 2023
STATUS
approved