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%I #16 Jun 05 2024 01:25:16
%S 0,0,1,1,8,3,136,29,928,942,66496,6175,2785856,205678,11122048
%N Half the number of permutations of 0..n with a zero n-th difference.
%e a(5)=3 because 0,1,2,3,4,5 ; 0,2,4,3,1,5 ; and 0,4,2,1,3,5 ; and their reverses have zero 5th differences.
%t (* naive and memory intensive implementation generating all permutations *)
%t Table[Count[
%t Map[Differences[#, n - 1] &,
%t Take[Permutations[Range[1, n]], Max[1, n!/2]]], {0}], {n, 1, 10}] (* _Olivier Gérard_, May 30 2024 *)
%o (Python)
%o from math import comb
%o from itertools import permutations
%o def A131502(n):
%o c = [-comb(n,i) if i&1 else comb(n,i) for i in range(n+1)]
%o return sum(1 for p in permutations(range(n+1)) if p[0]<p[-1] and not sum(c[i]*p[i] for i in range(n+1))) # _Chai Wah Wu_, Jun 04 2024
%K nonn,more
%O 0,5
%A _R. H. Hardin_, Aug 13 2007
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