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%I #28 Jun 03 2019 01:41:00
%S 1,3,16,115,1056,11774,154624,2337507,39984640,763546234,16101629952,
%T 371644257582,9319104528384,252270887452380,7332475985461248,
%U 227761317947788323,7529455986838732800,263948439074152148450
%N Normalized volume of center slice of n-dimensional cube: 2^n* n!*Vol({ (x_1,...,x_n) in [ 0,1 ]^n: n/2 <= Sum_{i = 1..n} x_i <= (n+1)/2 }).
%H D. Chakerian, D. Logothetti, <a href="http://dx.doi.org/10.2307/2690829 ">CubeSlices, Pictorial Triangles, and Probability</a>, Math. Mag., Vol. 64 (1991) 219-241.
%F V(d) = sum_{k=1}^{d-1} {d choose k-1} A_{d, k} where A_{k, d} denotes the Eulerian number (permutations of a d-set with k-1 descents) - see A008292.
%F Restated: a(n) = Sum_{k = 1..n} C(n,k-1)*A008292(n,k) for n>=1.
%F From _Peter Bala_, Jun 28 2016: (Start)
%F a(n) = 1/2*Sum_{k = 0..floor((n+1)/2)} (-1)^k*binomial(n + 1,k)*(n + 1 - 2*k)^n.
%F a(n) ~ sqrt(3)/2*(2/e)^(n+1)*(n+1)^n. (End)
%F a(2*n-1)/2^(2*n-2) = A025585(n) for n>=1. - _Peter Luschny_, Jun 30 2016
%p a := n -> add(binomial(n,k)*eulerian1(n,k), k=0..n-1):
%p seq(a(n), n=1..18); # _Peter Luschny_, Jun 30 2016
%t Eulerian1[n_, k_] = Sum[(-1)^j (k-j+1)^n Binomial[n+1, j], {j, 0, k+1}];
%t a[n_] := Sum[Binomial[n, k] Eulerian1[n, k], {k, 0, n-1}];
%t Array[a, 18] (* _Jean-François Alcover_, Jun 03 2019 *)
%Y Cf. A008292, A025585, A104098.
%K nonn,easy
%O 1,2
%A Guenter M. Ziegler (ziegler(AT)math.tu-berlin.de)
%E More terms from _Paul D. Hanna_, Mar 15 2006
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