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

1,2

Comments

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.

Links

Table of n, a(n) for n=1..64.

Formula

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).

Example

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.

Mathematica

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

Crossrefs

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.

Sequence in context: A078381 A118975 A215644 * A343503 A087724 A152047

Adjacent sequences: A288774 A288775 A288776 * A288778 A288779 A288780

Keyword

nonn,tabl,easy

Author

Omar E. Pol, Jun 15 2017

Status

approved