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

%I A193778 #36 Oct 07 2024 20:27:32
%S A193778 1,2,8,32,160,832,4864,29696,195584,1341440,9723904,73105408,
%T A193778 574062592,4657184768,39165624320,339133595648,3028204650496,
%U A193778 27760959422464,261439835078656,2521668748574720,24912764963127296,251559029812232192,2595790316527157248
%N A193778 Number of signed permutations of length 2n invariant under D and D'bar.
%C A193778 See section 6 in the Hardt reference.
%H A193778 Harvey P. Dale, <a href="/A193778/b193778.txt">Table of n, a(n) for n = 0..661</a>
%H A193778 Andy Hardt and Justin M. Troyka, <a href="http://puma.dimai.unifi.it/23_3/hardt_troyka.pdf">Restricted Symmetric Signed Permutations</a>, Department of Mathematics, Carleton College, 2013.
%F A193778 a(n) = 2*a(n-1) + 4*(n - 1)*a(n-2).
%F A193778 From _Peter Luschny_, May 30 2021: (Start)
%F A193778 a(n) = 2^n * hypergeom([-n/2, (1 - n)/2], [], 2).
%F A193778 a(n) = 2^n * A000085(n).
%F A193778 a(n) = n! * [x^n] exp(2*x*(x + 1)). (End)
%e A193778 For n = 1, the permutations of length 2 are (-1, 2) and (1, -2).
%p A193778 exp(2*x*(x + 1)): series(%, x, 23):
%p A193778 seq(n!*coeff(%, x, n), n = 0..22); # _Peter Luschny_, May 30 2021
%t A193778 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 *)
%t A193778 a[n_] := 2^n HypergeometricPFQ[{-n/2, (1-n)/2}, {}, 2];
%t A193778 Table[a[n], {n, 0, 22}] (* _Peter Luschny_, May 30 2021 *)
%o A193778 (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
%Y A193778 Column k=4 of A376826.
%Y A193778 Cf. A193777, A000085.
%K A193778 nonn
%O A193778 0,2
%A A193778 _Andy Hardt_, Aug 04 2011
%E A193778 Terms a(6) and beyond from _Andrew Howroyd_, Dec 08 2018