A363141 Expansion of g.f. A(x) satisfying 1/x = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 1, a(1) = 1.

1, 1, 0, 2, 3, 11, 23, 76, 188, 575, 1587, 4732, 13714, 40993, 121787, 367100, 1107371, 3367412, 10267404, 31468401, 96734992, 298488537, 923587457, 2866241029, 8916951360, 27808418089, 86910042122, 272180834822, 854004007736, 2684311988984, 8451232727631

Offset: 0

Author

Paul D. Hanna, Jun 09 2023

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^3 + 3*x^4 + 11*x^5 + 23*x^6 + 76*x^7 + 188*x^8 + 575*x^9 + 1587*x^10 + 4732*x^11 + 13714*x^12 + ...
		

Links

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

Crossrefs

Cf. A357227, A363312, A363313, A363314, A363315.

Programs

PARI

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.

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

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

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

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

a(n) ~ c * d^n / n^(3/2), where d = 3.3064656105288278... and c = 0.3845291573508... - Vaclav Kotesovec, Jun 09 2023