A288778 Triangle read by rows (1<=k<=n): T(n,k) = (n-k+1)*k! - (k-1)!

0, 1, 1, 2, 3, 4, 3, 5, 10, 18, 4, 7, 16, 42, 96, 5, 9, 22, 66, 216, 600, 6, 11, 28, 90, 336, 1320, 4320, 7, 13, 34, 114, 456, 2040, 9360, 35280, 8, 15, 40, 138, 576, 2760, 14400, 75600, 322560, 9, 17, 46, 162, 696, 3480, 19440, 115920, 685440, 3265920, 10, 19, 52, 186, 816, 4200, 24480, 156240, 1048320, 6894720, 36288000

Offset: 1

Author

Omar E. Pol, Jun 15 2017

Keywords

• nonn
• tabl
• easy

Comments

T(10,k) is also the number of positive integers with k digits in the sequence A215014. See Franklin T. Adams-Watters's comment in that entry. See also A288780.

Examples

			Triangle begins:
0;
1,   1;
2,   3,  4;
3,   5, 10,  18;
4,   7, 16,  42,  96;
5,   9, 22,  66, 216,  600;
6,  11, 28,  90, 336, 1320,  4320;
7,  13, 34, 114, 456, 2040,  9360,  35280;
8,  15, 40, 138, 576, 2760, 14400,  75600,  322560;
9,  17, 46, 162, 696, 3480, 19440, 115920,  685440, 3265920;
10, 19, 52, 186, 816, 4200, 24480, 156240, 1048320, 6894720, 36288000;
...
For n = 10 and k = 2; T(10,2) = 17 coincides with the number of positive terms with two digits in A215014 (see the first comment above).
		

Crossrefs

Column 1 gives A001477.

Row sums give A288780.

Cf. A000142, A004736, A166350, A215014, A288777.

Programs

Mathematica

Table[(n - k + 1) k! - (k - 1)!, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Jun 16 2017 *)

Formula

T(n,k) = A288777(n,k) - A000142(k-1), n>=1.