A390723 - OEIS

%I #10 Nov 19 2025 11:36:39

%S 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,

%T 1190,630,132,0,720,3528,8120,10290,7350,2772,429,0,5040,26136,65660,

%U 94766,82320,42504,12012,1430,0,40320,219168,590620,941976,942858,598752,234234,51480,4862

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

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

%e Triangle starts:

%e   [0]  1;

%e   [1]  0,     1;

%e   [2]  0,     1,      2;

%e   [3]  0,     2,      6,      5;

%e   [4]  0,     6,     22,     30,     14;

%e   [5]  0,    24,    100,    175,    140,     42;

%e   [6]  0,   120,    548,   1125,   1190,    630,    132;

%e   [7]  0,   720,   3528,   8120,  10290,   7350,   2772,    429;

%e   [8]  0,  5040,  26136,  65660,  94766,  82320,  42504,  12012,  1430;

%e   [9]  0, 40320, 219168, 590620, 941976, 942858, 598752, 234234, 51480, 4862;

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

%p A390723 := (n, k) -> (-1)^(n-k)*Stirling1(n, k)*CatalanNumber(k):

%p seq(seq(A390723(n, k), k = 0..n), n = 0..9);

%t 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 *)

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

%K nonn,tabl

%O 0,6

%A _Peter Luschny_, Nov 17 2025