A131502 Half the number of permutations of 1..(n+1) with a zero n-th difference.

0, 0, 1, 1, 8, 3, 136, 29, 928, 942, 66496, 6175, 2785856, 205678, 11122048, 134432251, 8816046848, 623303057, 906019171328

Offset

0,5

Links

Table of n, a(n) for n=0..18.

Example

a(5)=3 because 1,2,3,4,5,6 ; 1,3,5,4,2,6 ; and 1,5,3,2,4,6 ; and their reverses have zero 5th differences.

Mathematica

(* naive and memory intensive implementation generating all permutations *)
Table[Count[
  Map[Differences[#, n - 1] &,
   Take[Permutations[Range[1, n]], Max[1, n!/2]]], {0}], {n, 1, 10}] (* Olivier Gérard, May 30 2024 *)

Prog

(Python)
from math import comb
from itertools import permutations
def A131502(n):
    c = [-comb(n, i) if i&1 else comb(n, i) for i in range(n+1)]
    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

Crossrefs

Sequence in context: A038281 A302370 A303086 * A302891 A303626 A100253

Adjacent sequences: A131499 A131500 A131501 * A131503 A131504 A131505

Keyword

nonn,more

Author

R. H. Hardin, Aug 13 2007

Extensions

a(15)-a(18) from Bert Dobbelaere, Apr 14 2026

Status

approved