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%I #83 Feb 16 2023 05:07:19
%S 2,9,36,140,540,2079,8008,30888,119340,461890,1790712,6953544,
%T 27041560,105306075,410605200,1602881040,6263890380,24502865310,
%U 95937144600,375945078600,1474358525640,5786272150230,22724268808176,89301056353200,351140573438200,1381487341784004
%N a(n) = 6*(2*n+1)! / ((n!)^2*(n+3)).
%C If Y is a fixed 2-subset of a 2n-set X then a(n-2) is the number of (n-1)-subsets of X intersecting Y. - _Milan Janjic_, Oct 21 2007
%H G. C. Greubel, <a href="/A007946/b007946.txt">Table of n, a(n) for n = 0..1000</a>
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%F a(n) = C(2n+2, n) + C(2n+3, n). - _Emeric Deutsch_, May 16 2003
%F From _Karol A. Penson_, Aug 23 2014: (Start)
%F O.g.f.: ((-2+1/z^2-2/z)/sqrt(1-4*z)-1/z^2)/(2*z).
%F Representation as the n-th moment of a signed function: w(x) = sqrt(x/(4-x))*(x^2-2*x-2)/Pi on the segment x=(0,4). In Maple notation: a(n) = int(x^n*w(x), x=0..4). For x->0, w(x)->0, and for x->4, w(x)->infinity.
%F a(n) ~ (3/65536)*(4^n)*(-55332459+18443992*n - 6147840*n^2 + 2050048*n^3 - 688128*n^4 + 262144*n^5)/(n^(11/2)*sqrt(Pi)), for n->infinity.
%F (End)
%F a(n) = A001791(n+1) + A002054(n+1). - _Wesley Ivan Hurt_, Aug 23 2014
%F From _Peter Luschny_, Aug 25 2014: (Start)
%F a(n) = ((6*(2*n+1))/(n+3))* binomial(2*n,n).
%F a(n) has the asymptotic series 2^(2*n+3)*(1+(n+3)/((2*n+3))) *Sum_{k>=0}((num(k)/den(k))*(-n)^(-k))/sqrt(n*Pi). Here den(n) = 2^(4*n-A000120(n)) = A061549(n) and num(n) = 1, 25, 1297, 32755, 3249099, 79652055, 3876842453, 93900904955, 18138634602803, 437081823058595, 21036073578365391,... For example a(100) = 0.10602088220899083... *10^61 with the given values of num.
%F a(x) ~ exp(x*log(4)-(log(Pi)+cos(2*Pi*x)*(log(x) + 1/(4*x)))/2 + log((12*x+6)/ (3+x))). For example, this formula gives a(100) = 0.10602088... *10^61.
%F a(n) = A242986(2*n). (End)
%F a(n) = 12*4^n*Gamma(3/2+n)/(sqrt(Pi)*(3+n)*Gamma(1+n)). - _Peter Luschny_, Dec 14 2015
%F a(n) = 2*Sum_{i=0..n} (1/(i+1)*binomial(2*i+3,i+3)*binomial(2*(n-i),n-i)). - _Vladimir Kruchinin_, Apr 20 2016
%F E.g.f.: 2*(x*(-1 + 3*x)*BesselI(0,2*x) + (1 - 2*x + 3*x^2) * BesselI(1,2*x))*exp(2*x)/x^2. - _Ilya Gutkovskiy_, Apr 20 2016
%F D-finite with recurrence n*(n+3)*a(n) -2*(n+2)*(2*n+1)*a(n-1) = 0. - _R. J. Mathar_, Mar 30 2022
%F From _Amiram Eldar_, Feb 16 2023: (Start)
%F Sum_{n>=0} 1/a(n) = 8*Pi/(27*sqrt(3)) + 1/9.
%F Sum_{n>=0} (-1)^n/a(n) = 8*log(phi)/(5*sqrt(5)) + 1/15, where phi is the golden ratio (A001622). (End)
%p A007946:=n->binomial(2*n+2,n)+binomial(2*n+3,n): seq(A007946(n), n=0..30); # _Wesley Ivan Hurt_, Aug 23 2014
%t Table[Binomial[2 n + 2, n] + Binomial[2 n + 3, n], {n, 0, 30}] (* _Wesley Ivan Hurt_, Aug 23 2014 *)
%t Table[6*(2*n + 1)!/((n!)^2*(n + 3)), {n,0,50}] (* _G. C. Greubel_, Jan 23 2017 *)
%o (Magma) [Binomial(2*n+2, n) + Binomial(2*n+3, n) : n in [0..30]]; // _Wesley Ivan Hurt_, Aug 23 2014
%o (PARI) for(n=0,50, print1(6*(2*n + 1)!/((n!)^2*(n + 3)), ", ")) \\ _G. C. Greubel_, Jan 23 2017
%Y Cf. A000120, A001622, A001791, A002054, A061549, A242986.
%K nonn,easy
%O 0,1
%A _David W. Wilson_ and _Dean Hickerson_, Apr 21 1997
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