A395454 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(-j-1, -n-1)*E2(j, n-k), E2 the Eulerian numbers A201637.

1, 0, 0, 0, 2, 0, 0, 6, 2, 0, 0, 24, 34, 2, 0, 0, 120, 324, 98, 2, 0, 0, 720, 2988, 2096, 234, 2, 0, 0, 5040, 28944, 34704, 9804, 514, 2, 0, 0, 40320, 300816, 533592, 276004, 38856, 1082, 2, 0, 0, 362880, 3371040, 8137512, 6535272, 1766008, 139940, 2226, 2, 0

Offset

0,5

Links

Per Alexandersson, Rows 0..44 of triangle

Formula

Conjecture: Let P(n) be the generating polynomial (in t) of row n. Then P(n) = (n - 2*t + n*t - 2)*P(n-1) + (t - t^2)*P'(n-1) + t*(2*n-4)*P(n-2). - Per Alexandersson, Jun 27 2026

Example

Triangle starts:
  [0] 1;
  [1] 0,     0;
  [2] 0,     2,      0;
  [3] 0,     6,      2,      0;
  [4] 0,    24,     34,      2,      0;
  [5] 0,   120,    324,     98,      2,     0;
  [6] 0,   720,   2988,   2096,    234,     2,    0;
  [7] 0,  5040,  28944,  34704,   9804,   514,    2, 0;
  [8] 0, 40320, 300816, 533592, 276004, 38856, 1082, 2, 0;

Maple

A395454 := (n, k) -> local j; add(binomial(-j-1, -n-1)*combinat:-eulerian2(j, n-k), j=0..n):
seq(seq(A395454(n, k), k = 0..n), n = 0..8);

Crossrefs

Cf. A201637, A395452, A053871 (row sums).

Sequence in context: A345366 A278720 A221524 * A193009 A045866 A257549

Adjacent sequences: A395451 A395452 A395453 * A395455 A395456 A395457

Keyword

nonn,tabl

Author

Peter Luschny, Apr 23 2026

Status

approved