A025167 E.g.f: exp(x/(1-2*x))/(1-2*x).

1, 3, 17, 139, 1473, 19091, 291793, 5129307, 101817089, 2250495523, 54780588561, 1455367098923, 41888448785857, 1298019439099059, 43074477771208913, 1523746948247663611, 57229027745514785793, 2274027983943883110467

Offset: 0

Author

Wouter Meeussen

Keywords

• nonn

Comments

Polynomials in A021009 evaluated at -2.

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

Examples

			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
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..399
• 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. A025166, A025168, A002720.

Programs

Maple

a := n -> (-2)^n*KummerU(-n, 1, -1/2):
seq(simplify(a(n)), n=0..17); # Peter Luschny, Feb 12 2020

Mathematica

Table[ n! 2^n LaguerreL[ n, -1/2 ], {n, 0, 12} ]
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 *)
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 *)

PARI

{a(n)=n!^2*polcoeff(exp(2*x+x*O(x^n))*sum(m=0,n,x^m/m!^2),n)}

Formula

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]

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

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

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

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

a(n) = (-2)^n*KummerU(-n, 1, -1/2). - Peter Luschny, Feb 12 2020

Extensions

More terms from Vladeta Jovovic, Jan 29 2003