A363140 Expansion of g.f. A(x) satisfying 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(2*n))^(2*n+1).

1, 2, 5, 20, 86, 396, 1887, 9277, 46748, 240189, 1253474, 6625814, 35401302, 190878795, 1037296173, 5675580349, 31240459117, 172871809365, 961124621229, 5366264076784, 30076030970681, 169149177823245, 954301797559301, 5399467787889483, 30631118027908197

Offset: 0

Author

Paul D. Hanna, May 17 2023

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 20*x^3 + 86*x^4 + 396*x^5 + 1887*x^6 + 9277*x^7 + 46748*x^8 + 240189*x^9 + 1253474*x^10 + ...
		

Links

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

Crossrefs

Cf. A357232.

Programs

PARI

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

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

(2) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(4*n-3)) / (1 + A(x)*x^(2*n))^(2*n-1).