A390723 Triangle read by rows: T(n, k) = (-1)^(n-k)*Stirling1(n, k)*CatalanNumber(k).

1, 0, 1, 0, 1, 2, 0, 2, 6, 5, 0, 6, 22, 30, 14, 0, 24, 100, 175, 140, 42, 0, 120, 548, 1125, 1190, 630, 132, 0, 720, 3528, 8120, 10290, 7350, 2772, 429, 0, 5040, 26136, 65660, 94766, 82320, 42504, 12012, 1430, 0, 40320, 219168, 590620, 941976, 942858, 598752, 234234, 51480, 4862

Offset

0,6

Links

Table of n, a(n) for n=0..54.

Formula

T(n, k) = (-1)^(n-k)*A132393(n, k)*A000108(k).

Example

Triangle starts:
  [0]  1;
  [1]  0,     1;
  [2]  0,     1,      2;
  [3]  0,     2,      6,      5;
  [4]  0,     6,     22,     30,     14;
  [5]  0,    24,    100,    175,    140,     42;
  [6]  0,   120,    548,   1125,   1190,    630,    132;
  [7]  0,   720,   3528,   8120,  10290,   7350,   2772,    429;
  [8]  0,  5040,  26136,  65660,  94766,  82320,  42504,  12012,  1430;
  [9]  0, 40320, 219168, 590620, 941976, 942858, 598752, 234234, 51480, 4862;

Maple

CatalanNumber := n -> binomial(2*n, n)/(n + 1):
A390723 := (n, k) -> (-1)^(n-k)*Stirling1(n, k)*CatalanNumber(k):
seq(seq(A390723(n, k), k = 0..n), n = 0..9);

Mathematica

T[n_, k_]:=(-1)^(n-k)*StirlingS1[n, k]*CatalanNumber[k]; Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* James C. McMahon, Nov 19 2025 *)

Crossrefs

Cf. A132393, A000108, A086662 (row sums), A086672 (alternating row sums), A390724.

Sequence in context: A327116 A157491 A094385 * A355260 A291799 A295027

Adjacent sequences: A390720 A390721 A390722 * A390724 A390725 A390726

Keyword

nonn,tabl

Author

Peter Luschny, Nov 17 2025

Status

approved