A361766 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n/A(-x))^(n+2).

1, 1, 2, 5, 12, 27, 57, 123, 280, 666, 1614, 3955, 9733, 23949, 58967, 145844, 363137, 910339, 2295192, 5811070, 14754567, 37542078, 95715596, 244567665, 626388406, 1608131393, 4137707994, 10667045757, 27546269363, 71241831762, 184508259405, 478501423792

Offset: 0

Author

Paul D. Hanna, Mar 26 2023

Keywords

• nonn

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds for all y as a formal power series in x.

Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n) * (y - x^n)^(n+2), which holds for all y as a formal power series in x. - Paul D. Hanna, Jan 19 2025

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 27*x^5 + 57*x^6 + 123*x^7 + 280*x^8 + 666*x^9 + 1614*x^10 + 3955*x^11 + 9733*x^12 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A358952, A355866.

Programs

PARI

{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = -polcoeff( sum(m=-#A,#A, (-x)^m * (1 - (-x)^m/Ser(A))^(m+2) ), #A-3));A[n+1]}
for(n=0,35,print1(a(n),", "))

Formula

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

(1) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) - (-x)^n)^(n+2) / A(x)^n.

(2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)) * A(x)^n / (1 - (-x)^n*A(x))^(n-2).

From Paul D. Hanna, Jan 19 2025: (Start)

(3) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (1 - (-x)^n)^n * A(x)^n.

(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)) / ((1 - (-x)^n)^n * A(x)^n).

(End)

a(n) ~ c * d^n / n^(3/2), where d = 2.71312501383... and c = 3.43853109... - Vaclav Kotesovec, Mar 31 2023