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A335823 Triangle read by rows: A080779 with rows reversed.

%I A335823 #29 Jul 01 2020 20:31:33
%S A335823 1,1,1,2,3,1,6,12,6,0,24,60,40,0,-4,120,360,300,0,-60,0,720,2520,2520,
%T A335823 0,-840,0,120,5040,20160,23520,0,-11760,0,3360,0,40320,181440,241920,
%U A335823 0,-169344,0,80640,0,-12096,362880,1814400,2721600,0,-2540160,0,1814400,0,-544320,0
%N A335823 Triangle read by rows: A080779 with rows reversed.
%F A335823 T(n,k) = n!*(n-1)*(n-2)*...*(n-k+1)*(-1)^k*Bk/k! where Bk is a Bernoulli number and T(n,0) = (n-1)! and T(n,m) = 0 if m >= n.
%e A335823 Triangle begins:
%e A335823    1;
%e A335823    1,  1;
%e A335823    2,  3,  1;
%e A335823    6, 12,  6, 0;
%e A335823   24, 60, 40, 0, -4;
%e A335823   ...
%t A335823 Table[If[k == 0, (n - 1)!, n!*Product[n - j, {j, k - 1}]*(-1)^k*BernoulliB[k]/k!], {n, 10}, {k, 0, n - 1}] // Flatten (* _Michael De Vlieger_, Jun 27 2020 *)
%o A335823 (PARI) T(n,k) = if (k==0, (n-1)!, n!*prod(j=1,k-1, n-j)*(-1)^k*bernfrac(k)/k!);
%o A335823 tabl(nn) = for(n=1, nn, for (k=0, n-1, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, Jun 25 2020
%Y A335823 Cf. A000142 (row sums).
%Y A335823 Cf. A080779 (same triangle with rows reversed).
%Y A335823 Cf. A027641/A027642 (Bernoulli numbers).
%K A335823 sign,tabl
%O A335823 1,4
%A A335823 _John O. Oladokun_, Jun 25 2020