%I #46 Nov 10 2023 12:27:43
%S 1,1,2,3,7,13,34,73,209,501,1546,4051,13327,37633,130922,394353,
%T 1441729,4596553,17572114,58941091,234662231,824073141,3405357682,
%U 12470162233,53334454417,202976401213,896324308634,3535017524403,16083557845279,65573803186921
%N Denominators of continued fraction for alternating factorial.
%C Starting (1, 2, 3, ...) with offset 0 = eigensequence of an infinite lower triangular matrix with 1's in the main diagonal and the natural numbers repeated in the subdiagonal. - _Gary W. Adamson_, Feb 14 2011
%C a(n) is the number of involutions of [n] such that every 2-cycle contains one odd and one even element; a(4) = 7: 1234, 1243, 1324, 2134, 2143, 4231, 4321. - _Alois P. Heinz_, Feb 14 2013
%H Alois P. Heinz, <a href="/A056953/b056953.txt">Table of n, a(n) for n = 0..400</a>
%H Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, <a href="https://arxiv.org/abs/2210.17461">Ramified inverse and planar monoids</a>, arXiv:2210.17461 [math.RT], 2022.
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F a(0)=1; a(1)=1; a(n) = a(n-1) + n*a(n-2)/2.
%F a(n) = Sum_{k=0..[n/2]} k!*C([n/2],k)*C([(n+1)/2],k). - _Paul D. Hanna_, Oct 31 2006
%F a(n) ~ n^(n/2 + 1/4) / (2^(n/2 + 3/4) * exp(n/2 - sqrt(2*n) + 1/2)) * (1 + (25 + 6*(-1)^n)/(24*sqrt(2*n)) + (397 + 156*(-1)^n)/(2304*n)). - _Vaclav Kotesovec_, Feb 22 2019
%p a:= proc(n) option remember; `if`(n<4, [1, 1, 2, 3][n+1],
%p ((4*n-2)*a(n-2) +2*a(n-3) -(n-2)*(n-3)*a(n-4)) /4)
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Feb 14 2013
%t Table[Sum[k!*Binomial[Floor[n/2], k]*Binomial[Floor[(n+1)/2], k] , {k,0,Floor[n/2]}], {n,0,30}] (* _G. C. Greubel_, May 16 2018 *)
%o (PARI) a(n)=sum(k=0,n\2,k!*binomial(n\2,k)*binomial((n+1)\2,k)) \\ _Paul D. Hanna_, Oct 31 2006
%o (Magma) [(&+[Factorial(k)*Binomial(Floor(n/2),k)*Binomial(Floor((n+1)/2),k): k in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, May 16 2018
%Y Bisections are A000262 and A002720.
%Y Cf. A124428, diagonals of A088699.
%K nonn,easy
%O 0,3
%A _Aleksandar Petojevic_, Sep 05 2000