A083886 Expansion of e.g.f. exp(3*x)*exp(x^2).

1, 3, 11, 45, 201, 963, 4899, 26253, 147345, 862083, 5238459, 32957037, 214117209, 1433320515, 9867008979, 69734001357, 505212273441, 3747124863747, 28418591888235, 220152270759597, 1740363304031721, 14027180742479043, 115176800996769411, 962726355659386125, 8186311912829551281, 70769800810139187843

Offset: 0

Author

Paul Barry, May 09 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A000898.

Hankel transform is A108400. - Paul Barry, Jun 13 2009

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

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

Examples

			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
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..717
• Aram Bingham and Ozlem Ugurlu, Sects and lattice paths over the Lagrangian Grassmannian, arXiv:1903.07229 [math.CO], 2019.
• A. Hardt and J. M. Troyka, Restricted symmetric signed permutations, Pure Mathematics and Applications, Vol. 23 (No. 3, 2012), pp. 179--217.
• A. Hardt and J. M. Troyka, Slides (associated with the Hardt and Troyka reference above).

Crossrefs

Cf. A001813, A047974.

Programs

Mathematica

a = {1, 3}; For[n = 2, n < 26, n++, a = Append[a, 3 a[[n]] + 2 (n - 1) a[[n - 1]]]]; a (* Justin M. Troyka, Aug 05 2011 *)
CoefficientList[Series[Exp[3*x+x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 25 2013 *)
Table[Abs[HermiteH[n, 3 I/2]], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 11 2016 *)

PARI

my(x='x+O('x^26)); Vec(serlaplace(exp(3*x)*exp(x^2))) /* Joerg Arndt, Jul 12 2012 */

Formula

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

From Paul Barry, Jun 13 2009: (Start)

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

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

a(n) = i^n*Hermite_H(n, -3i/2), i=sqrt(-1). - Paul Barry, Jun 15 2009

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

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

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

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