A373422 Triangle read by rows: T(n,k) = number of permutations of [n] starting from k that have zero (n-1)-th differences. (n>=1, 1<=k<=n).

0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 4, 2, 4, 2, 4, 3, 0, 0, 0, 0, 3, 40, 36, 40, 40, 40, 36, 40, 29, 0, 0, 0, 0, 0, 0, 29, 232, 152, 240, 200, 208, 200, 240, 152, 232, 235, 142, 140, 257, 168, 168, 257, 140, 142, 235, 11712, 13216, 12208, 12384, 11408, 11136, 11408, 12384, 12208, 13216, 11712

Offset: 1

Author

Seiichi Manyama, Jun 04 2024

Keywords

• nonn
• tabl

Examples

			T(3,1) = 1 because [1,2,3] have zero 2nd differences.
  1 2 3
   1 1
    0
Triangle starts:
    0;
    0,   0;
    1,   0,   1;
    1,   0,   0,   1;
    4,   2,   4,   2,   4;
    3,   0,   0,   0,   0,   3;
   40,  36,  40,  40,  40,  36,  40;
   29,   0,   0,   0,   0,   0,   0,  29;
  232, 152, 240, 200, 208, 200, 240, 152, 232;
  235, 142, 140, 257, 168, 168, 257, 140, 142, 235;
		

Links

• Seiichi Manyama, Rows n = 1..14, flattened

Crossrefs

Row sums give 2 * A131502(n-1).

Programs

PARI

tabl(n) = my(nn=vector(n)); forperm([1..n], p, if(sum(k=1, n, (-1)^k*binomial(n-1, k-1)*p[k])==0, nn[p[1]]++)); nn;

Formula

T(n,k) = T(n,n+1-k) for 1<=k<=n.

If p is prime, T(p+1,k) = 0 for 2 <= k <= p.