A288780 -id:A288780 - cp's OEIS frontend

A288777 Triangle read by rows in which column k lists the positive multiples of the factorial of k, with 1 <= k <= n.

1, 2, 2, 3, 4, 6, 4, 6, 12, 24, 5, 8, 18, 48, 120, 6, 10, 24, 72, 240, 720, 7, 12, 30, 96, 360, 1440, 5040, 8, 14, 36, 120, 480, 2160, 10080, 40320, 9, 16, 42, 144, 600, 2880, 15120, 80640, 362880, 10, 18, 48, 168, 720, 3600, 20160, 120960, 725760, 3628800, 11, 20, 54, 192, 840, 4320, 25200, 161280, 1088640

Offset: 1

Omar E. Pol, Jun 15 2017

T(n,k) is the number of k-digit numbers in base n+1 with distinct positive digits that form an integer interval when sorted.
T(9,k) is also the number of numbers with k digits in A288528.
The number of terms in A288528 is also A014145(9) = 462331, the same as the sum of the 9th row of this triangle.
Removing the left column of A137267 and of A137948 then this triangle appears in both cases.

			Triangle begins:
   1;
   2,  2;
   3,  4,  6;
   4,  6, 12,  24;
   5,  8, 18,  48, 120;
   6, 10, 24,  72, 240,  720;
   7, 12, 30,  96, 360, 1440,  5040;
   8, 14, 36, 120, 480, 2160, 10080,  40320;
   9, 16, 42, 144, 600, 2880, 15120,  80640,  362880;
  10, 18, 48, 168, 720, 3600, 20160, 120960,  725760, 3628800;
  11, 20, 54, 192, 840, 4320, 25200, 161280, 1088640, 7257600, 39916800;
  ...
For n = 9 and k = 2: T(9,2) is the number of numbers with two digits in A288528.
For n = 9 the row sum is 9 + 16 + 42 + 144 + 600 + 2880 + 15120 + 80640 + 362880 = 462331, the same as A014145(9) and also the same as the number of terms in A288528.

Right border gives A000142, n>=1.
Middle diagonal gives A001563, n>=1.
Row sums give A014145, n>=1.
Column 1..4: A000027, A005843, A008588, A008606.
Cf. A007489, A000142, A004736, A137267, A137948, A166350, A288528, A288778, A288780.

Table[(n - k + 1) k!, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Jun 15 2017 *)

T(n,k) = (n-k+1)*k! = (n-k+1)*A000142(k) = A004736(n,k)*A166350(n,k).
T(n,k) = Sum_{j=1..n} A166350(j,k).
T(n,k) =  A288778(n,k) + A000142(k-1).

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

Original entry on oeis.org

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

Omar E. Pol, Jun 15 2017

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.

			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).

Column 1 gives A001477.
Row sums give A288780.
Cf. A000142, A004736, A166350, A215014, A288777.

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

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

cp's OEIS Frontend

A288777 Triangle read by rows in which column k lists the positive multiples of the factorial of k, with 1 <= k <= n.

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