A193778 Number of signed permutations of length 2n invariant under D and D'bar.

1, 2, 8, 32, 160, 832, 4864, 29696, 195584, 1341440, 9723904, 73105408, 574062592, 4657184768, 39165624320, 339133595648, 3028204650496, 27760959422464, 261439835078656, 2521668748574720, 24912764963127296, 251559029812232192, 2595790316527157248

Offset: 0

Author

Andy Hardt, Aug 04 2011

Keywords

• nonn

Comments

See section 6 in the Hardt reference.

Examples

			For n = 1, the permutations of length 2 are (-1, 2) and (1, -2).
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..661
• Andy Hardt and Justin M. Troyka, Restricted Symmetric Signed Permutations, Department of Mathematics, Carleton College, 2013.

Crossrefs

Column k=4 of A376826.

Cf. A193777, A000085.

Programs

Maple

exp(2*x*(x + 1)): series(%, x, 23):
seq(n!*coeff(%, x, n), n = 0..22); # Peter Luschny, May 30 2021

Mathematica

RecurrenceTable[{a[0]==1,a[1]==2,a[n]==2a[n-1]+4(n-1)a[n-2]},a,{n,30}] (* Harvey P. Dale, Mar 14 2021 *)
a[n_] := 2^n HypergeometricPFQ[{-n/2, (1-n)/2}, {}, 2];
Table[a[n], {n, 0, 22}] (* Peter Luschny, May 30 2021 *)

PARI

seq(n)={my(v=vector(n+1)); v[1]=1; v[2]=2; for(n=2, n, v[n+1] = 2*v[n] + 4*(n-1)*v[n-1]); v} \\ Andrew Howroyd, Dec 08 2018

Formula

a(n) = 2*a(n-1) + 4*(n - 1)*a(n-2).

From Peter Luschny, May 30 2021: (Start)

a(n) = 2^n * hypergeom([-n/2, (1 - n)/2], [], 2).

a(n) = 2^n * A000085(n).

a(n) = n! * [x^n] exp(2*x*(x + 1)). (End)

Extensions

Terms a(6) and beyond from Andrew Howroyd, Dec 08 2018