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

%I A202828 #20 Aug 28 2025 00:45:01
%S A202828 1,4,36,400,5776,97344,1915456,42406144,1049760000,28558296064,
%T A202828 848579961856,27271456395264,943132599095296,34877026635366400,
%U A202828 1373536895379849216,57351382681767706624,2530646978003730497536,117614221470591038521344,5742190572014854792806400
%N A202828 Expansion of e.g.f.: exp(4*x/(1-2*x)) / sqrt(1-4*x^2).
%C A202828 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
%H A202828 G. C. Greubel, <a href="/A202828/b202828.txt">Table of n, a(n) for n = 0..350</a>
%H A202828 Lực Ta, <a href="https://arxiv.org/abs/2508.16772">Good involutions of twisted conjugation subquandles and Alexander quandles</a>, arXiv:2508.16772 [math.GT], 2025.
%F A202828 a(n) = A000898(n)^2, where the e.g.f. of A000898 is exp(2*x + x^2).
%F A202828 a(n) = ( Sum_{k=0..[n/2]} 2^(n-2*k) * n!/((n-2*k)!*k!) )^2.
%F A202828 a(n) = ( Sum_{k=0..n} Stirling1(n, k)*2^k*Bell(k) )^2. [From formula by Vladeta Jovovic in A000898].
%F A202828 a(n) ~ n^n*exp(2*sqrt(2*n)-1-n)*2^(n-1). - _Vaclav Kotesovec_, May 23 2013
%F A202828 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
%F A202828 a(n) = 2^n*A277378(n). - _R. J. Mathar_, Jan 20 2020
%e A202828 E.g.f.: A(x) = 1 + 4*x + 36*x^2/3! + 400*x^3/3! + 5776*x^4/4! + 97344*x^5/5! +...
%e A202828 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! +...
%t A202828 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 *)
%o A202828 (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)}
%o A202828 (PARI) {a(n)=sum(k=0,n\2,2^(n-2*k)*n!/((n-2*k)!*k!))^2}
%o A202828 (PARI)
%o A202828 {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o A202828 {Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
%o A202828 {a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k)*2^k)^2}
%o A202828 (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(4*x/(1-2*x))/Sqrt(1-4*x^2) ))); // _G. C. Greubel_, Jun 21 2022
%o A202828 (SageMath)
%o A202828 def A202828_list(prec):
%o A202828     P.<x> = PowerSeriesRing(QQ, prec)
%o A202828     return P( exp(4*x/(1-2*x))/sqrt(1-4*x^2) ).egf_to_ogf().list()
%o A202828 A202828_list(40) # _G. C. Greubel_, Jun 21 2022
%Y A202828 Cf. A000898, A202827, A202829, A202831, A202833, A202835, A202836.
%Y A202828 Cf. A387317.
%K A202828 nonn,changed
%O A202828 0,2
%A A202828 _Paul D. Hanna_, Dec 25 2011