A363144 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(4*n-1))^(n+1).

1, 1, 1, 1, 3, 7, 13, 21, 35, 64, 125, 243, 459, 852, 1593, 3035, 5857, 11326, 21835, 42053, 81246, 157741, 307421, 600207, 1172805, 2294197, 4495735, 8827574, 17363422, 34198201, 67429181, 133097669, 263028031, 520406201, 1030749582, 2043553947, 4055171751

Offset: 0

Author

Paul D. Hanna, May 17 2023

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 3*x^4 + 7*x^5 + 13*x^6 + 21*x^7 + 35*x^8 + 64*x^9 + 125*x^10 + 243*x^11 + 459*x^12 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A359711, A363142, A363143.

Programs

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 * (Ser(A) + x^(4*m-1))^(m+1) ),#A-1));A[n+1]}
for(n=0,40,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.

(1) 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(4*n-1))^(n+1).

(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(4*n*(n-1)) / (1 + A(x)*x^(4*n+1))^(n-1).

(3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(4*n-1))^n.

(4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(5*n) * (A(x) + x^(4*n-1))^(n-1).

(5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(4*n^2) / (1 + A(x)*x^(4*n+1))^n.