A357791 a(n) = coefficient of x^n in A(x) such that: x = Sum_{n=-oo..+oo} x^n * (1 - x^n * A(-x)^n)^n.

1, 1, 2, 5, 21, 88, 377, 1654, 7424, 34000, 158274, 746525, 3559456, 17128250, 83078147, 405754479, 1993777057, 9849668910, 48892589632, 243739139810, 1219789105228, 6125813250402, 30862120708266, 155937956267432, 790019313067409, 4012282344217699, 20423575546661000

Offset: 0

Author

Paul D. Hanna, Dec 24 2022

Keywords

• nonn

Comments

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

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 21*x^4 + 88*x^5 + 377*x^6 + 1654*x^7 + 7424*x^8 + 34000*x^9 + 158274*x^10 + 746525*x^11 + 3559456*x^12 + ...
SPECIFIC VALUES.
A(x) = 3/2 at x = 0.1850570503493984408934312903280642188437354418734...
A(1/6) = 1.3085832721715442420948608003299892250459754159045...
		

Links

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

Crossrefs

Cf. A357399, A359672.

Programs

PARI

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

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

a(n) ~ c * d^n / n^(3/2), where d = 5.390297559554269719991046... and c = 0.267652299887938085649... - Vaclav Kotesovec, Dec 25 2022