A288778 -id:A288778 - cp's OEIS frontend

A288780 Zero together with the row sums of A288778.

0, 0, 2, 9, 36, 165, 918, 6111, 47304, 416097, 4091130, 44417043, 527456556, 6798432069, 94499679582, 1408924024695, 22425642181008, 379514672913321, 6804212771165634, 128827325000617947, 2568509718703606260, 53787877376348226573, 1180349932648067726886

Offset: 0

Omar E. Pol, Jun 15 2017

nonn

For n >= 2, a(n) is the number of numbers in base n with consecutive digits after reordering.

a(10) = 4091130 is also the number of positive terms in the finite sequence A215014, hence a(10) + 1 = 4091131 is the total number of terms in that sequence.

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A215014, A288777, A288778.

a:= proc(n) option remember; `if`(n<3, n*(n-1),
      n*(a(n-1)*n/(n-1)-a(n-2)*(n-1)/(n-2)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 16 2017

{0}~Join~Map[Total, Table[(n - k + 1) k! - (k - 1)!, {n, 22}, {k, n}]] (* Michael De Vlieger, Jun 21 2017 *)

More terms from Alois P. Heinz, Jun 16 2017

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A288780 Zero together with the row sums of A288778.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions