A335823 Triangle read by rows: A080779 with rows reversed.

1, 1, 1, 2, 3, 1, 6, 12, 6, 0, 24, 60, 40, 0, -4, 120, 360, 300, 0, -60, 0, 720, 2520, 2520, 0, -840, 0, 120, 5040, 20160, 23520, 0, -11760, 0, 3360, 0, 40320, 181440, 241920, 0, -169344, 0, 80640, 0, -12096, 362880, 1814400, 2721600, 0, -2540160, 0, 1814400, 0, -544320, 0

Offset: 1

Author

John O. Oladokun, Jun 25 2020

Keywords

• sign
• tabl

Examples

			Triangle begins:
   1;
   1,  1;
   2,  3,  1;
   6, 12,  6, 0;
  24, 60, 40, 0, -4;
  ...
		

Crossrefs

Cf. A000142 (row sums).

Cf. A080779 (same triangle with rows reversed).

Cf. A027641/A027642 (Bernoulli numbers).

Programs

Mathematica

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 *)

PARI

T(n,k) = if (k==0, (n-1)!, n!*prod(j=1,k-1, n-j)*(-1)^k*bernfrac(k)/k!);
tabl(nn) = for(n=1, nn, for (k=0, n-1, print1(T(n,k), ", ")); print); \\ Michel Marcus, Jun 25 2020

Formula

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.