A363139 Expansion of A(x) satisfying -x = Sum_{n=-oo..+oo} (-x)^n * (1 - (-x)^n)^n / A(x)^n.

1, 1, 2, 3, 10, 29, 72, 190, 520, 1413, 3888, 10839, 30421, 86218, 246499, 708931, 2050584, 5962100, 17407554, 51019081, 150052163, 442677295, 1309668356, 3884884796, 11551622175, 34425468793, 102807253860, 307617338332, 922112808168, 2768808168311, 8327028966970

Offset: 0

Author

Paul D. Hanna, May 30 2023

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 + 3*x^3 + 10*x^4 + 29*x^5 + 72*x^6 + 190*x^7 + 520*x^8 + 1413*x^9 + 3888*x^10 + 10839*x^11 + 30421*x^12 + ...
SPECIFIC VALUES.
G.f. A(x) diverges at x = 1/3.
A(1/sqrt(10)) = 2.740968311596221258712215041101550216...
A(3/10) = 2.04409403049365965943794935957987166879615299154...
A(x) = 2 at x = 0.29764678443183662600376771573865711430158997980267844885...
A(1/4) = 1.54451964019778087973376938515481313055726531377...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A357399.

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 +x*O(x^#A))^n / Ser(A)^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)^n / A(x)^n.

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