A390725 - OEIS

%I #13 Nov 19 2025 11:45:59

%S 1,0,1,0,2,2,0,6,12,5,0,24,72,60,14,0,120,480,600,280,42,0,720,3600,

%T 6000,4200,1260,132,0,5040,30240,63000,58800,26460,5544,429,0,40320,

%U 282240,705600,823200,493920,155232,24024,1430

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

%H Vincenzo Librandi, <a href="/A390725/b390725.txt">Table of n, a(n) for n = 0..6104</a>

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

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

%e Triangle begins:

%e   [0]  1;

%e   [1]  0,     1;

%e   [2]  0,     2,      2;

%e   [3]  0,     6,     12,      5;

%e   [4]  0,    24,     72,     60,     14;

%e   [5]  0,   120,    480,    600,    280,     42;

%e   [6]  0,   720,   3600,   6000,   4200,   1260,    132;

%e   [7]  0,  5040,  30240,  63000,  58800,  26460,   5544,   429;

%e   [8]  0, 40320, 282240, 705600, 823200, 493920, 155232, 24024, 1430;

%p T := (n, k) -> binomial(n - 1, k - 1)*binomial(2*k, k)*n!/(k+1)!:

%p seq(seq(T(n, k), k=0..n), n=0..9);

%t 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 *)

%o (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;

%o 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

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

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Nov 17 2025