A083886 - OEIS

%I #61 Oct 29 2019 08:58:48

%S 1,3,11,45,201,963,4899,26253,147345,862083,5238459,32957037,

%T 214117209,1433320515,9867008979,69734001357,505212273441,

%U 3747124863747,28418591888235,220152270759597,1740363304031721,14027180742479043,115176800996769411,962726355659386125,8186311912829551281,70769800810139187843

%N Expansion of e.g.f. exp(3*x)*exp(x^2).

%C Binomial transform of A000898.

%C Hankel transform is A108400. - _Paul Barry_, Jun 13 2009

%C a(n) is the number of self-inverse 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.

%C a(n) is also the number of skew-symmetric (n,n)-clans, or the number of B-orbits in the symmetric space of type CI, Sp_{2n}(C)/GL_n(C) where B is a Borel subgroup of Sp_{2n}(C). - _Aram Bingham_, Oct 08 2019

%H Michael De Vlieger, <a href="/A083886/b083886.txt">Table of n, a(n) for n = 0..717</a>

%H Aram Bingham, Ozlem Ugurlu, <a href="https://arxiv.org/abs/1903.07229">Sects and lattice paths over the Lagrangian Grassmannian</a>, arXiv:1903.07229 [math.CO], 2019.

%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 E.g.f.: exp(3*x+x^2).

%F From _Paul Barry_, Jun 13 2009: (Start)

%F G.f.: 1/(1-3x-2x^2/(1-3x-4x^2/(1-3x-6x^2/(1-3x-8x^2/(1-... (continued fraction);

%F a(n) = Sum_{k=0..floor(n/2)} C(n,2k) * (2k)! * 3^(n-2k) / k!. (End)

%F a(n) = i^n*Hermite_H(n, -3i/2), i=sqrt(-1). - _Paul Barry_, Jun 15 2009

%F a(0) = 1; a(1) = 3; a(n) = 3*a(n-1) + 2*(n-1)*a(n-2) for n >= 2. - _Justin M. Troyka_, Aug 05 2011

%F E.g.f. 1 + (x+3)*x/(G(0)-x^2-3*x) where G(k)= x^2 + 3*x + k + 1 - (x+3)*x*(k+1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - _Sergei N. Gladkovskii_, Jul 12 2012

%F G.f.: 1/Q(0) where Q(k) = 1 + 2*x*k - 2*x - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 07 2013

%F a(n) ~ n^(n/2)*2^(n/2-1/2)*exp(3*sqrt(n/2)-n/2-9/8) * (1+21*sqrt(2)/(32*sqrt(n))). - _Vaclav Kotesovec_, Jun 25 2013

%e Since a(2) = 11, there are 11 self-inverse signed permutations of 4 that are equal to their reverse-complements and avoid (-2,-1). Some of these are: (+3,+4,+1,+2), (+4,-2,-3,+1), (-1,+3,+2,-4), (-1,-2,-3,-4). - _Justin M. Troyka_, Aug 05 2011

%t a = {1, 3}; For[n = 2, n < 13, n++, a = Append[a, 3 a[[n]] + 2 (n - 1) a[[n - 1]]]]; a // _Justin M. Troyka_, Aug 05 2011.

%t CoefficientList[Series[Exp[3*x+x^2], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 25 2013 *)

%t Table[Abs[HermiteH[n, 3 I/2]], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 11 2016 *)

%o (PARI)  x='x+O('x^66);  Vec(serlaplace(exp(3*x)*exp(x^2))) /* _Joerg Arndt_, Jul 12 2012 */

%Y Cf. A001813, A047974.

%K easy,nonn

%O 0,2

%A _Paul Barry_, May 09 2003