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A242986 a(n) = 6*(n+1)!/((3+floor(n/2))*(floor(n/2)!)^2).

%I A242986 #21 Feb 17 2023 10:06:34
%S A242986 2,4,9,36,36,216,140,1120,540,5400,2079,24948,8008,112112,30888,
%T A242986 494208,119340,2148120,461890,9237800,1790712,39395664,6953544,
%U A242986 166885056,27041560,703080560,105306075,2948570100,410605200,12318156000,1602881040,51292193280,6263890380
%N A242986 a(n) = 6*(n+1)!/((3+floor(n/2))*(floor(n/2)!)^2).
%F A242986 a(n) = (6*(n+1)/(3+floor(n/2)))*A056040(n).
%F A242986 a(2*n) = A007946(n).
%F A242986 Recurrence: a(n) = a(n-1)*(n+1)*(4*(n+4))/(n^2*(n+6)) if n mod 2 = 0 else a(n-1)*(n+1) for n>0, a(0) = 2.
%F A242986 Asymptotic: a(x) ~ exp(x*log(2) - log(Pi)/2 - cos(Pi*x)*(log(x/2) + 1/(2*x))/2 + log(6*(x+1)) - log(3+floor(x/2))) for x>=1.
%F A242986 G.f.: (4*x-1)/(2*x^6) + (-16*x^7+16*x^6-48*x^5+12*x^4+48*x^3-12*x^2-8*x+2)/(4*(1-4*x^2)^(3/2)*x^6). - _Robert Israel_, Aug 25 2014
%F A242986 Sum_{n>=0} 1/a(n) = Pi^2/54 + 19*Pi/(54*sqrt(3)) + 1/9. - _Amiram Eldar_, Feb 17 2023
%p A242986 A056040 := n -> n!/iquo(n,2)!^2;
%p A242986 A242986 := n -> (6*(n+1)/(3+iquo(n,2)))*A056040(n);
%p A242986 seq(A242986(n), n=0..32);
%t A242986 Table[6(n + 1)!/((3 + Floor[n/2])*(Floor[n/2]!)^2), {n, 0, 30}] (* _Wesley Ivan Hurt_, Aug 26 2014 *)
%o A242986 (Sage)
%o A242986 @CachedFunction
%o A242986 def A242986(n):
%o A242986     if n == 0: return 2
%o A242986     h = (n+1)*A242986(n-1)
%o A242986     if 2.divides(n):
%o A242986         h *= (4*(n+4))/(n^2*(n+6))
%o A242986     return h
%o A242986 [A242986(n) for n in range(33)]
%o A242986 (Magma) [6*Factorial(n+1)/((3+Floor(n/2))*(Factorial(Floor(n/2)))^2) : n in [0..30]]; // _Wesley Ivan Hurt_, Aug 26 2014
%Y A242986 Cf. A007946, A056040.
%K A242986 nonn
%O A242986 0,1
%A A242986 _Peter Luschny_, Aug 25 2014