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A240558 a(n) = 2^n*n!/((floor(n/2)+1)*floor(n/2)!^2).

%I A240558 #27 Mar 01 2020 07:45:17
%S A240558 1,2,4,24,32,320,320,4480,3584,64512,43008,946176,540672,14057472,
%T A240558 7028736,210862080,93716480,3186360320,1274544128,48432676864,
%U A240558 17611882496,739699064832,246566354944,11342052327424,3489862254592,174493112729600,49855175065600
%N A240558 a(n) = 2^n*n!/((floor(n/2)+1)*floor(n/2)!^2).
%H A240558 G. C. Greubel, <a href="/A240558/b240558.txt">Table of n, a(n) for n = 0..1000</a>
%F A240558 O.g.f.: ((i*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2), where i=sqrt(-1).
%F A240558 For a recurrence see the Sage program.
%F A240558 a(n) = 2^n*A057977(n)
%F A240558 a(2*k) = A151403(k) = 2^k*A151374(k) = 4^k*A000108(k).
%F A240558 a(2*k+1) = A099045(k+1) = 2^k*A069723(k+2) = 4^k*A000984(k+1).
%F A240558 From _Peter Luschny_, Jan 31 2015: (Start)
%F A240558 a(n) = Sum_{k=0..n} A056040(n)*C(n,k)/(floor(n/2)+1).
%F A240558 a(n) = Sum_{k=0..n} n!*C(n,k)/((floor(n/2)+1)*(floor(n/2)!)^2).
%F A240558 a(n) = 2^n*n!*[x^n]((x+1)*hypergeom([],[2],x^2)).
%F A240558 a(n) ~ 2^(n+N)/((n+1)^<n>*sqrt(Pi*(2*N+1))); here <n> = 1 if n is even, 0 otherwise and N = n+<n>+1. (End)
%F A240558 Conjecture: -(n+2)*(n^2-5)*a(n) +8*(-2*n-1)*a(n-1) +16*(n-1)*(n^2+2*n-4)*a(n-2)=0. - _R. J. Mathar_, Jun 14 2016
%p A240558 A240558 := n -> 2^n*n!/((iquo(n,2)+1)*iquo(n,2)!^2):
%p A240558 seq(A240558(n), n=0..30);
%t A240558 Table[SeriesCoefficient[((I*(2*x*(8*x+1)-1))/Sqrt[16*x^2-1]-2*x+1) /(8*x^2), {x,0,n}], {n,0,22}]
%o A240558 (Sage)
%o A240558 def A240558():
%o A240558     x, n = 1, 1
%o A240558     while True:
%o A240558         yield x
%o A240558         m = 2*n if is_odd(n) else 8/(n+2)
%o A240558         x *= m
%o A240558         n += 1
%o A240558 a = A240558(); [next(a) for i in range(36)]
%o A240558 (PARI) x='x+O('x^50); Vec(round((I*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2)) \\ _G. C. Greubel_, Apr 05 2017
%Y A240558 Cf. A000108, A000984, A056040, A057977, A069723, A099045, A151374, A151403.
%K A240558 nonn,easy
%O A240558 0,2
%A A240558 _Peter Luschny_, Apr 14 2014