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%I #18 Nov 23 2019 21:06:50
%S 0,2,12,120,1320,17856,273056,4772624,92626944,1986317024,46556867456,
%T 1184827221584,32524270418432,958020105786536
%N Sum of absolute values of n-th differences over all permutations of {0, 1, ..., n}.
%C a(n) <= ((n+1)! - 2*A131502(n))*A130783(n).
%C Every term is even because the n-th difference of a permutation and its reversal are the same up to sign.
%e For n = 2, the second differences of the (2+1)! = 6 permutations of {0,1,2} are:
%e [0,1,2] -> [1, 1] -> 0,
%e [0,2,1] -> [2,-1] -> -3,
%e [1,0,2] -> [-1, 2] -> 3,
%e [1,2,0] -> [1,-2] -> -3,
%e [2,0,1] -> [-2, 1] -> 3, and
%e [2,1,0] -> [-1,-1] -> 0.
%e The sum of the absolute values of these second differences is 0 + 3 + 3 + 3 + 3 + 0 = 12.
%t a[n_] := Block[{x, k}, k = CoefficientList[(x - 1)^n, x]; Sum[Abs[k.p], {p, Permutations@ Range[0, n]}]]; Array[a, 10, 0] (* _Giovanni Resta_, Nov 23 2019 *)
%Y Cf. A130783, A131502, A327845.
%K nonn,more
%O 0,2
%A _Peter Kagey_, Nov 22 2019
%E a(10) from _Alois P. Heinz_, Nov 22 2019
%E a(11)-a(13) from _Giovanni Resta_, Nov 23 2019
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