A379765 G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + (-x)^n)^(2*n-1).

1, 4, 16, 108, 764, 5772, 45608, 372112, 3110868, 26511720, 229465696, 2011560120, 17823251908, 159361875452, 1436070211128, 13029220181024, 118919107720504, 1091130632899108, 10058749510188900, 93119868866604632, 865350260237277984, 8069341311245971172, 75482617925071807900

Offset: 0

Author

Paul D. Hanna, Jan 22 2025

Keywords

• nonn

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n) * (y - x^n)^(n+1), which holds for all y as a formal power series in x.

Examples

			G.f.: A(x) = 1 + 4*x + 16*x^2 + 108*x^3 + 764*x^4 + 5772*x^5 + 45608*x^6 + 372112*x^7 + 3110868*x^8 + 26511720*x^9 + 229465696*x^10 + ...
SPECIFIC VALUES.
A(t) = 2 at t = 0.09653361905915170411984272932017391451579223633328063596...
A(t) = 3/2 at t = 0.075349068495101237164879111573650564301786145264462860...
A(t) = 4/3 at t = 0.059270539047675430011940298576693796977637316335946787...
A(t) = 5/4 at t = 0.048395689470367112406846758430274833454550275773025303...
A(1/11) = 1.7859531236503504891314901027866679467841212478816...
A(1/12) = 1.6197331904782587957364168880788048296765610762165...
A(1/13) = 1.5208322951137414977098221192098933277226208754204...
A(1/14) = 1.4524825965114467294893588185221903177323240271997...
A(1/20) = 1.2612955244894656446063844025636148211057476645966...
		

Links

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

Crossrefs

Cf. A379763.

Programs

PARI

{a(n) = my(V=[1]); for(i=1,n, V=concat(V,0); A = Ser(V);
V[#V] = polcoef(-2 + 4*sum(n=-#V,#V, (-1)^n * x^(2*n) * (A + (-x)^n)^(2*n-1) ),#V-1) );V[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) 1/2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + (-x)^n)^(2*n-1).

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

a(n) ~ c * d^n / n^(3/2), where d = 10.0028789085783346757183844487387995725278... and c = 0.82085228741035116338050876364461533993... - Vaclav Kotesovec, Jan 22 2025