A363143 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(3*n-1))^(n+1).

1, 1, 1, 3, 7, 13, 25, 52, 111, 235, 495, 1054, 2271, 4923, 10703, 23354, 51190, 112668, 248783, 550875, 1223107, 2722766, 6075619, 13586390, 30442616, 68339788, 153683822, 346173172, 780948750, 1764312745, 3991321375, 9040912764, 20503640896, 46552634034

Offset: 0

Author

Paul D. Hanna, May 17 2023

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 7*x^4 + 13*x^5 + 25*x^6 + 52*x^7 + 111*x^8 + 235*x^9 + 495*x^10 + 1054*x^11 + 2271*x^12 + ...
		

Links

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

Crossrefs

Cf. A359711, A363142, A363144.

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^(3*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^(3*n-1))^(n+1).

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

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

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

(5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(3*n^2) / (1 + A(x)*x^(3*n+1))^n.