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A355996 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) = n!/(k! * floor(n/k)!).

%I A355996 #14 Jul 22 2022 10:36:21
%S A355996 1,1,1,1,3,1,1,6,4,1,1,30,20,5,1,1,60,60,30,6,1,1,420,420,210,42,7,1,
%T A355996 1,840,3360,840,336,56,8,1,1,7560,10080,7560,3024,504,72,9,1,1,15120,
%U A355996 100800,75600,15120,5040,720,90,10,1,1,166320,1108800,831600,166320,55440,7920,990,110,11,1
%N A355996 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) = n!/(k! * floor(n/k)!).
%F A355996 E.g.f. of column k: (1 - x^k) * (exp(x^k) - 1)/(k! * (1 - x)).
%e A355996 Triangle begins:
%e A355996   1;
%e A355996   1,   1;
%e A355996   1,   3,    1;
%e A355996   1,   6,    4,   1;
%e A355996   1,  30,   20,   5,   1;
%e A355996   1,  60,   60,  30,   6,  1;
%e A355996   1, 420,  420, 210,  42,  7, 1;
%e A355996   1, 840, 3360, 840, 336, 56, 8, 1;
%e A355996   ...
%t A355996 T[n_, k_] := n!/(k!*Floor[n/k]!); Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Jul 22 2022 *)
%o A355996 (PARI) T(n, k) = n!/(k!*(n\k)!);
%Y A355996 Row sums give A355991.
%Y A355996 Column k=1..3 give A000012, A355989, A355990.
%K A355996 nonn,tabl
%O A355996 1,5
%A A355996 _Seiichi Manyama_, Jul 22 2022