This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A288777 #63 Jun 25 2017 15:11:20
%S A288777 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 A288777 360,1440,5040,8,14,36,120,480,2160,10080,40320,9,16,42,144,600,2880,
%U A288777 15120,80640,362880,10,18,48,168,720,3600,20160,120960,725760,3628800,11,20,54,192,840,4320,25200,161280,1088640
%N A288777 Triangle read by rows in which column k lists the positive multiples of the factorial of k, with 1 <= k <= n.
%C A288777 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 A288777 T(9,k) is also the number of numbers with k digits in A288528.
%C A288777 The number of terms in A288528 is also A014145(9) = 462331, the same as the sum of the 9th row of this triangle.
%C A288777 Removing the left column of A137267 and of A137948 then this triangle appears in both cases.
%F A288777 T(n,k) = (n-k+1)*k! = (n-k+1)*A000142(k) = A004736(n,k)*A166350(n,k).
%F A288777 T(n,k) = Sum_{j=1..n} A166350(j,k).
%F A288777 T(n,k) = A288778(n,k) + A000142(k-1).
%e A288777 Triangle begins:
%e A288777 1;
%e A288777 2, 2;
%e A288777 3, 4, 6;
%e A288777 4, 6, 12, 24;
%e A288777 5, 8, 18, 48, 120;
%e A288777 6, 10, 24, 72, 240, 720;
%e A288777 7, 12, 30, 96, 360, 1440, 5040;
%e A288777 8, 14, 36, 120, 480, 2160, 10080, 40320;
%e A288777 9, 16, 42, 144, 600, 2880, 15120, 80640, 362880;
%e A288777 10, 18, 48, 168, 720, 3600, 20160, 120960, 725760, 3628800;
%e A288777 11, 20, 54, 192, 840, 4320, 25200, 161280, 1088640, 7257600, 39916800;
%e A288777 ...
%e A288777 For n = 9 and k = 2: T(9,2) is the number of numbers with two digits in A288528.
%e A288777 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 A288777 Table[(n - k + 1) k!, {n, 11}, {k, n}] // Flatten (* _Michael De Vlieger_, Jun 15 2017 *)
%Y A288777 Right border gives A000142, n>=1.
%Y A288777 Middle diagonal gives A001563, n>=1.
%Y A288777 Row sums give A014145, n>=1.
%Y A288777 Column 1..4: A000027, A005843, A008588, A008606.
%Y A288777 Cf. A007489, A000142, A004736, A137267, A137948, A166350, A288528, A288778, A288780.
%K A288777 nonn,tabl,easy
%O A288777 1,2
%A A288777 _Omar E. Pol_, Jun 15 2017