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A288777
Triangle read by rows in which column k lists the positive multiples of the factorial of k, with 1 <= k <= n.
3
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.
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.
Sequence in context: A078381 A118975 A215644 * A343503 A087724 A152047
KEYWORD
nonn,tabl,easy
AUTHOR
Omar E. Pol, Jun 15 2017
STATUS
approved