A361779 Expansion of g.f. A(x) satisfying 1/x = Sum_{n=-oo..+oo} x^n * (x^(2*n) - (-1)^n*A(x))^(n+1).

1, 1, 2, 5, 10, 21, 51, 121, 282, 688, 1704, 4212, 10528, 26626, 67630, 172590, 443156, 1143034, 2958829, 7687875, 20043717, 52410511, 137417383, 361225349, 951755240, 2513057208, 6648904064, 17624116631, 46796906873, 124460500129, 331517863145, 884305712723, 2362007410465

Offset: 0

Author

Paul D. Hanna, May 10 2023

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 21*x^5 + 51*x^6 + 121*x^7 + 282*x^8 + 688*x^9 + 1704*x^10 + 4212*x^11 + 10528*x^12 + ...
SPECIFIC VALUES.
A(1/4) = 1.54381930928063102950885404708273996504264975892127868985...
A(3/10) = 1.8845579890166759655973763714847523770459496427989251...
A(1/3) = 2.35223102094304184442834405817178151095013948472323960819...
		

Links

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

Crossrefs

Cf. A361778.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, x^m * (x^(2*m) - (-1)^m*Ser(A))^(m+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 may be defined by the following.

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

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

a(n) ~ c * d^n / n^(3/2), where d = 2.791690127253271... and c = 2.581668816660... - Vaclav Kotesovec, May 11 2023