A081114 Triangle read by rows: T(n,k) = n*T(n-1,k) + n - k starting at T(n,n)=0.

0, 1, 0, 4, 1, 0, 15, 5, 1, 0, 64, 23, 6, 1, 0, 325, 119, 33, 7, 1, 0, 1956, 719, 202, 45, 8, 1, 0, 13699, 5039, 1419, 319, 59, 9, 1, 0, 109600, 40319, 11358, 2557, 476, 75, 10, 1, 0, 986409, 362879, 102229, 23019, 4289, 679, 93, 11, 1, 0, 9864100, 3628799, 1022298, 230197, 42896, 6795, 934, 113, 12, 1, 0

Offset: 0

Author

Henry Bottomley, Apr 16 2003

Keywords

• nonn
• tabl

Comments

Taking the triangle into negative values of n and k would produce results close to (k+1)*e*n! - 1, i.e., one less than multiples of A000522 for nonnegative n.

Examples

			Triangle begins
    0;
    1,   0;
    4,   1,  0;
   15,   5,  1, 0;
   64,  23,  6, 1, 0;
  325, 119, 33, 7, 1, 0;
		

Crossrefs

Columns include A007526 and A033312.

Programs

PARI

T(n,k) = if (k==n, 0, n*T(n-1,k) + n - k);
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n, k), ", ");); print(););} \\ Michel Marcus, Jun 16 2019

Formula

For k > 0, T(n, k) = ceiling((A001339(k-1)/(k-1)! - (k-1)*e) *n! - 1) where A001339(k-1) = ceiling((k-1)!*(k-1)*e) for k > 1.

T(n, 0) = floor(e*n! - 1) for n > 0; T(n, 1) = n! - 1. T(n, n)=0; T(n, n-1) = n+2; T(n, n-2) = n^2 + 3*n + 5 = A027688(n+1).

Extensions

More terms from Michel Marcus, Jun 16 2019