A359922 a(n) = coefficient of x^n in A(x) where x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^n * (2 + x^n*A(x)^n)^n.

1, 1, 4, 9, 42, 187, 775, 3470, 16085, 76521, 368274, 1791494, 8829531, 43964379, 220667042, 1115235384, 5671532510, 29004157940, 149056379047, 769368598912, 3986831368824, 20733495321171, 108175116519808, 566067951728994, 2970221822319878, 15624080964153005

Offset: 0

Author

Paul D. Hanna, Jan 18 2023

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 42*x^4 + 187*x^5 + 775*x^6 + 3470*x^7 + 16085*x^8 + 76521*x^9 + 368274*x^10 + ...
where
x = ... + x^6*A(x)^9/(1 + 2*x^3*A(x)^3)^3 - x^2*A(x)^4/(1 + 2*x^2*A(x)^2)^2 + A(x)/(1 + 2*x*A(x)) - 1 + x*(2 + x*A(x)) - x^2*(2 + x^2*A(x)^2)^2 + x^3*(2 + x^3*A(x)^3)^3 + ... + (-1)^(n-1) * x^n * (2 + x^n*A(x)^n)^n + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A359672, A359923.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(x - sum(m=-#A, #A, (-1)^(m-1) * x^m * (2 + (x*Ser(A))^m)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

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

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

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