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A011818
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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 }).
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3
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1, 3, 16, 115, 1056, 11774, 154624, 2337507, 39984640, 763546234, 16101629952, 371644257582, 9319104528384, 252270887452380, 7332475985461248, 227761317947788323, 7529455986838732800, 263948439074152148450
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OFFSET
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1,2
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LINKS
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FORMULA
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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.
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)
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MAPLE
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a := n -> add(binomial(n, k)*eulerian1(n, k), k=0..n-1):
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MATHEMATICA
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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}];
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Guenter M. Ziegler (ziegler(AT)math.tu-berlin.de)
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EXTENSIONS
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STATUS
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approved
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