A322255 Triangle T(n,k) giving the number of permutations of 1..n with no adjacent elements within k in value, for n >= 2, 1 <= k <= floor(n/2).
2, 6, 24, 2, 120, 14, 720, 90, 2, 5040, 646, 32, 40320, 5242, 368, 2, 362880, 47622, 3984, 72, 3628800, 479306, 44304, 1496, 2, 39916800, 5296790, 521606, 25384, 160, 479001600, 63779034, 6564318, 399848, 6056, 2, 6227020800, 831283558, 88422296, 6231544, 161136, 352
Offset: 2
Examples
Irregular triangle starts: n\k| 1 2 3 4 5 ---+--------------------------------- 2 | 2; 3 | 6; 4 | 24, 2; 5 | 120, 14; 6 | 720, 90, 2; 7 | 5040, 646, 32; 8 | 40320, 5242, 368, 2; 9 | 362880, 47622, 3984, 72; 10 | 3628800, 479306, 44304, 1496, 2;
Formula
T(n,k) = Sum_{j=k..floor(n/2)} A129534(n,j). - Alois P. Heinz, May 20 2023