A288777 - OEIS

%I #63 Jun 25 2017 15:11:20

%S 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,

%T 360,1440,5040,8,14,36,120,480,2160,10080,40320,9,16,42,144,600,2880,

%U 15120,80640,362880,10,18,48,168,720,3600,20160,120960,725760,3628800,11,20,54,192,840,4320,25200,161280,1088640

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

%C 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.

%C T(9,k) is also the number of numbers with k digits in A288528.

%C The number of terms in A288528 is also A014145(9) = 462331, the same as the sum of the 9th row of this triangle.

%C Removing the left column of A137267 and of A137948 then this triangle appears in both cases.

%F T(n,k) = (n-k+1)*k! = (n-k+1)*A000142(k) = A004736(n,k)*A166350(n,k).

%F T(n,k) = Sum_{j=1..n} A166350(j,k).

%F T(n,k) =  A288778(n,k) + A000142(k-1).

%e Triangle begins:

%e    1;

%e    2,  2;

%e    3,  4,  6;

%e    4,  6, 12,  24;

%e    5,  8, 18,  48, 120;

%e    6, 10, 24,  72, 240,  720;

%e    7, 12, 30,  96, 360, 1440,  5040;

%e    8, 14, 36, 120, 480, 2160, 10080,  40320;

%e    9, 16, 42, 144, 600, 2880, 15120,  80640,  362880;

%e   10, 18, 48, 168, 720, 3600, 20160, 120960,  725760, 3628800;

%e   11, 20, 54, 192, 840, 4320, 25200, 161280, 1088640, 7257600, 39916800;

%e   ...

%e For n = 9 and k = 2: T(9,2) is the number of numbers with two digits in A288528.

%e 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.

%t Table[(n - k + 1) k!, {n, 11}, {k, n}] // Flatten (* _Michael De Vlieger_, Jun 15 2017 *)

%Y Right border gives A000142, n>=1.

%Y Middle diagonal gives A001563, n>=1.

%Y Row sums give A014145, n>=1.

%Y Column 1..4: A000027, A005843, A008588, A008606.

%Y Cf. A007489, A000142, A004736, A137267, A137948, A166350, A288528, A288778, A288780.

%K nonn,tabl,easy

%O 1,2

%A _Omar E. Pol_, Jun 15 2017