A390724 Triangle read by rows: T(n, k) = Stirling2(n, k)*CatalanNumber(k).

1, 0, 1, 0, 1, 2, 0, 1, 6, 5, 0, 1, 14, 30, 14, 0, 1, 30, 125, 140, 42, 0, 1, 62, 450, 910, 630, 132, 0, 1, 126, 1505, 4900, 5880, 2772, 429, 0, 1, 254, 4830, 23814, 44100, 35112, 12012, 1430, 0, 1, 510, 15125, 108780, 291942, 349272, 198198, 51480, 4862

Offset

0,6

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..6104 (first 110 rows of the triangle, flattened)

Formula

T(n, k) = A048993(n, k)*A000108(k).

Example

Triangle begins:
  [0]  1;
  [1]  0, 1;
  [2]  0, 1,   2;
  [3]  0, 1,   6,     5;
  [4]  0, 1,  14,    30,     14;
  [5]  0, 1,  30,   125,    140,     42;
  [6]  0, 1,  62,   450,    910,    630,    132;
  [7]  0, 1, 126,  1505,   4900,   5880,   2772,    429;
  [8]  0, 1, 254,  4830,  23814,  44100,  35112,  12012,  1430;
  [9]  0, 1, 510, 15125, 108780, 291942, 349272, 198198, 51480, 4862;

Maple

CatalanNumber := n -> binomial(2*n, n)/(n + 1):
A390724 := (n, k) -> Stirling2(n, k)*CatalanNumber(k):
seq(seq(A390724(n, k), k = 0..n), n = 0..9);

Mathematica

T[n_, k_]:=StirlingS2[n, k]*CatalanNumber[k]; Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* James C. McMahon, Nov 18 2025 *)

Prog

(Magma) T := func<n, k | StirlingSecond(n, k) * Catalan(k)>;
vals := []; for n in [0..9] do for k in [0..n] do Append(~vals, T(n, k)); end for; end for; vals; // Vincenzo Librandi, Nov 19 2025

Crossrefs

Cf. A048993, A000108, A064856 (row sums), A355290 (alternating row sums), A390723.

Sequence in context: A114709 A293147 A331047 * A264550 A089949 A085845

Adjacent sequences: A390721 A390722 A390723 * A390725 A390726 A390727

Keyword

nonn,tabl

Author

Peter Luschny, Nov 17 2025

Status

approved