A385909 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (x^(2*n) - A(x))^(3*n+1).

1, 1, 3, 9, 31, 122, 493, 2086, 9106, 40764, 186206, 865068, 4076020, 19437711, 93655043, 455293416, 2230636436, 11003483165, 54607084364, 272453502850, 1365876088389, 6876896373019, 34757806185051, 176291771193079, 897001780346928, 4577362669389502, 23420275560794225, 120123996076924029

Offset: 0

Author

Paul D. Hanna, Jul 29 2025

Keywords

• nonn

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(2*n+1))^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 31*x^4 + 122*x^5 + 493*x^6 + 2086*x^7 + 9106*x^8 + 40764*x^9 + 186206*x^10 + 865068*x^11 + 4076020*x^12 + ...
		

Links

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

Crossrefs

Cf. A355866.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, x^n*(x^(2*n) - Ser(A))^(3*n+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 formulas.

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

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

a(n) ~ c * d^n / n^(3/2), where d = 5.437310827623... and c = 0.230314472... - Vaclav Kotesovec, Aug 04 2025