A011818 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 }).

1, 3, 16, 115, 1056, 11774, 154624, 2337507, 39984640, 763546234, 16101629952, 371644257582, 9319104528384, 252270887452380, 7332475985461248, 227761317947788323, 7529455986838732800, 263948439074152148450

Offset: 1

Author

Guenter M. Ziegler (ziegler(AT)math.tu-berlin.de)

Keywords

• nonn
• easy

Links

• D. Chakerian, D. Logothetti, CubeSlices, Pictorial Triangles, and Probability, Math. Mag., Vol. 64 (1991) 219-241.

Crossrefs

Cf. A008292, A025585, A104098.

Programs

Maple

a := n -> add(binomial(n,k)*eulerian1(n,k), k=0..n-1):
seq(a(n), n=1..18); # Peter Luschny, Jun 30 2016

Mathematica

Eulerian1[n_, k_] = Sum[(-1)^j (k-j+1)^n Binomial[n+1, j], {j, 0, k+1}];
a[n_] := Sum[Binomial[n, k] Eulerian1[n, k], {k, 0, n-1}];
Array[a, 18] (* Jean-François Alcover, Jun 03 2019 *)

Formula

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.

Restated: a(n) = Sum_{k = 1..n} C(n,k-1)*A008292(n,k) for n>=1.

From Peter Bala, Jun 28 2016: (Start)

a(n) = 1/2*Sum_{k = 0..floor((n+1)/2)} (-1)^k*binomial(n + 1,k)*(n + 1 - 2*k)^n.

a(n) ~ sqrt(3)/2*(2/e)^(n+1)*(n+1)^n. (End)

a(2*n-1)/2^(2*n-2) = A025585(n) for n>=1. - Peter Luschny, Jun 30 2016

Extensions

More terms from Paul D. Hanna, Mar 15 2006