%I #52 Feb 02 2021 11:55:11
%S 1,3,17,139,1473,19091,291793,5129307,101817089,2250495523,
%T 54780588561,1455367098923,41888448785857,1298019439099059,
%U 43074477771208913,1523746948247663611,57229027745514785793,2274027983943883110467
%N E.g.f: exp(x/(1-2*x))/(1-2*x).
%C Polynomials in A021009 evaluated at -2.
%C Also, a(n) is the number of signed permutations of length 2n that are equal to their reverse-complements and avoid the pattern (-2,-1). As a result, a(n) also gives the same thing but for avoiding any one of (-1,-2), (+2,+1) or (+1,+2) instead of (-2,-1) (see the Hardt and Troyka reference). - _Justin M. Troyka_, Aug 05 2011
%H Seiichi Manyama, <a href="/A025167/b025167.txt">Table of n, a(n) for n = 0..399</a>
%H A. Hardt and J. M. Troyka, <a href="http://www.mat.unisi.it/newsito/puma/public_html/23_3/hardt_troyka.pdf">Restricted symmetric signed permutations</a>, Pure Mathematics and Applications, Vol. 23 (No. 3, 2012), pp. 179--217.
%H A. Hardt and J. M. Troyka, <a href="https://apps.carleton.edu/curricular/math/assets/Andy_hardt_slides.pdf">Slides</a> (associated with the Hardt and Troyka reference above).
%F a(n) = Sum_{k=0..n} k!*2^k*binomial(n, k)^2. - _Robert G. Wilson v_, Mar 16 2005 [corrected by _Ilya Gutkovskiy_, Oct 01 2018]
%F a(n) = Sum_{k=0..n-1} 2^{n-1-k}*[(n-1)! ]^2/[(k!)^2*(n-1-k)! ]. - Huajun Huang (huanghu(AT)auburn.edu), Oct 10 2005
%F a(0) = 1; a(1) = 3; a(n) = (4n-1) * a(n-1) - 4 (n-1)^2 * a(n-2) for n >= 2. - _Justin M. Troyka_, Aug 05 2011
%F E.g.f.: exp(2*x) * Sum_{n>=0} x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. - _Paul D. Hanna_, Nov 18 2011
%F a(n) ~ n^(n+1/4)*2^(n-1/4)*exp(-n+sqrt(2*n)-1/4) * (1 + sqrt(2)/(3*sqrt(n))). - _Vaclav Kotesovec_, Jun 22 2013
%F a(n) = (-2)^n*KummerU(-n, 1, -1/2). - _Peter Luschny_, Feb 12 2020
%e Since a(2) = 17, there are 17 signed permutations of 4 that are equal to their reverse-complements and avoid (-2,-1). Some of these are: (+1,+3,+2,+4), (+2,-1,-4,+3), (+3,-1,-4,+2), (-1,-2,-3,-4). - _Justin M. Troyka_, Aug 05 2011
%p a := n -> (-2)^n*KummerU(-n, 1, -1/2):
%p seq(simplify(a(n)), n=0..17); # _Peter Luschny_, Feb 12 2020
%t Table[ n! 2^n LaguerreL[ n, -1/2 ], {n, 0, 12} ]
%t f[n_] := Sum[k!*2^k*Binomial[n, k]^2, {k, 0, n}]; Table[ f[n], {n, 0, 17}] (* _Robert G. Wilson v_, Mar 16 2005 *)
%t a = {1, 3}; For[n = 2, n < 13, n++, a = Append[a, (4 n - 1) a[[n]] - 4 (n - 1)^2 a[[n - 1]]]]; a (* _Justin M. Troyka_, Aug 05 2011 *)
%o (PARI) {a(n)=n!^2*polcoeff(exp(2*x+x*O(x^n))*sum(m=0,n,x^m/m!^2),n)}
%Y Cf. A025166, A025168, A002720.
%K nonn
%O 0,2
%A _Wouter Meeussen_
%E More terms from _Vladeta Jovovic_, Jan 29 2003