A390725 Triangle read by rows: T(n, k) = Lah(n, k)*CatalanNumber(k), and Lah = A271703.

1, 0, 1, 0, 2, 2, 0, 6, 12, 5, 0, 24, 72, 60, 14, 0, 120, 480, 600, 280, 42, 0, 720, 3600, 6000, 4200, 1260, 132, 0, 5040, 30240, 63000, 58800, 26460, 5544, 429, 0, 40320, 282240, 705600, 823200, 493920, 155232, 24024, 1430

Offset

0,5

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..6104

Formula

T(n, k) = A271703(n, k)*A000108(k).

T(n + 1, n) = A005430(n) (Apéry numbers).

Example

Triangle begins:
  [0]  1;
  [1]  0,     1;
  [2]  0,     2,      2;
  [3]  0,     6,     12,      5;
  [4]  0,    24,     72,     60,     14;
  [5]  0,   120,    480,    600,    280,     42;
  [6]  0,   720,   3600,   6000,   4200,   1260,    132;
  [7]  0,  5040,  30240,  63000,  58800,  26460,   5544,   429;
  [8]  0, 40320, 282240, 705600, 823200, 493920, 155232, 24024, 1430;

Maple

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

Mathematica

T[n_, k_] := Binomial[n-1, k-1] Binomial[2 k, k] n!/(k+1)!; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* Vincenzo Librandi, Nov 19 2025 *)

Prog

(Magma) T:= function(n, k) if n eq 0 and k eq 0 then return 1; else return Binomial(n-1, k-1)*Binomial(2*k, k)*Factorial(n)/Factorial(k+1); end if; end function;
seq := []; for n in [0..9] do for k in [0..n] do Append(~seq, T(n, k)); end for; end for; seq; // Vincenzo Librandi, Nov 19 2025

Crossrefs

Cf. A271703, A000108, A005430, A317276 (row sums), A390723, A390724.

Sequence in context: A323777 A292317 A285675 * A361488 A219859 A366230

Adjacent sequences: A390722 A390723 A390724 * A390726 A390727 A390728

Keyword

nonn,tabl

Author

Peter Luschny, Nov 17 2025

Status

approved