A202828 Expansion of e.g.f.: exp(4*x/(1-2*x)) / sqrt(1-4*x^2).

1, 4, 36, 400, 5776, 97344, 1915456, 42406144, 1049760000, 28558296064, 848579961856, 27271456395264, 943132599095296, 34877026635366400, 1373536895379849216, 57351382681767706624, 2530646978003730497536, 117614221470591038521344, 5742190572014854792806400

Offset: 0

Author

Paul D. Hanna, Dec 25 2011

Keywords

• nonn
• changed

Comments

a(n) is the number of good involutions of the linear Alexander quandle (Z/4nZ, 2n+1); see Ta, Thm. 5.11 and cf. A387317. - Luc Ta, Aug 26 2025

Examples

			E.g.f.: A(x) = 1 + 4*x + 36*x^2/3! + 400*x^3/3! + 5776*x^4/4! + 97344*x^5/5! +...
where A(x) = 1 + 2^2*x + 6^2*x^2/2! + 20^2*x^3/3! + 76^2*x^4/4! + 312^2*x^5/5! +...+ A000898(n)^2*x^n/n! +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..350
• Lực Ta, Good involutions of twisted conjugation subquandles and Alexander quandles, arXiv:2508.16772 [math.GT], 2025.

Crossrefs

Cf. A000898, A202827, A202829, A202831, A202833, A202835, A202836.

Cf. A387317.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(4*x/(1-2*x))/Sqrt(1-4*x^2) ))); // G. C. Greubel, Jun 21 2022

Mathematica

CoefficientList[Series[Exp[4*x/(1-2*x)]/Sqrt[1-4*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

PARI

{a(n)=n!*polcoeff(exp(4*x/(1-2*x)+x*O(x^n))/sqrt(1-4*x^2+x*O(x^n)),n)}

PARI

{a(n)=sum(k=0,n\2,2^(n-2*k)*n!/((n-2*k)!*k!))^2}

PARI

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k)*2^k)^2}

SageMath

def A202828_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-2*x))/sqrt(1-4*x^2) ).egf_to_ogf().list()
A202828_list(40) # G. C. Greubel, Jun 21 2022

Formula

a(n) = A000898(n)^2, where the e.g.f. of A000898 is exp(2*x + x^2).

a(n) = ( Sum_{k=0..[n/2]} 2^(n-2*k) * n!/((n-2*k)!*k!) )^2.

a(n) = ( Sum_{k=0..n} Stirling1(n, k)*2^k*Bell(k) )^2. [From formula by Vladeta Jovovic in A000898].

a(n) ~ n^n*exp(2*sqrt(2*n)-1-n)*2^(n-1). - Vaclav Kotesovec, May 23 2013

D-finite with recurrence: a(n) = 2*(n+1)*a(n-1) + 4*(n-1)*(n+1)*a(n-2) - 8*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

a(n) = 2^n*A277378(n). - R. J. Mathar, Jan 20 2020