A334318 Number T(n,k) of integers in base n having exactly k distinct digits such that the number formed by the consecutive subsequence of the initial j digits is divisible by j for all j in {1,...,k}; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 2, 1, 3, 1, 0, 4, 5, 5, 2, 5, 6, 6, 1, 0, 6, 13, 18, 8, 7, 2, 7, 15, 33, 34, 16, 7, 0, 8, 25, 50, 58, 52, 21, 8, 3, 9, 28, 67, 98, 101, 57, 30, 7, 0, 10, 41, 115, 168, 220, 88, 51, 9, 4, 1, 11, 45, 134, 275, 398, 315, 220, 126, 32, 10, 0, 12, 61, 206, 428, 690, 568, 503, 158, 32, 5, 1, 0

Offset

1,2

Links

Alois P. Heinz, Rows n = 1..25, flattened

Example

T(4,3) = 5: 102, 120, 201, 123, 321 (written in base 4):
T(7,2) = 15: 13, 15, 20, 24, 26, 31, 35, 40, 42, 46, 51, 53, 60, 62, 64 (written in base 7)
T(10,1) = 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
T(10,10) = 1: 3816547290.
Triangle T(n,k) begins:
   1;
   2,  1;
   3,  1,   0;
   4,  5,   5,   2;
   5,  6,   6,   1,   0;
   6, 13,  18,   8,   7,   2;
   7, 15,  33,  34,  16,   7,   0;
   8, 25,  50,  58,  52,  21,   8,   3;
   9, 28,  67,  98, 101,  57,  30,   7,  0;
  10, 41, 115, 168, 220,  88,  51,   9,  4,  1;
  11, 45, 134, 275, 398, 315, 220, 126, 32, 10, 0;
  12, 61, 206, 428, 690, 568, 503, 158, 32,  5, 1, 0;
  ...

Maple

b:= proc(n, s, w) option remember; `if`(s={}, 0, (k-> add((t->
      `if`(t=0, x, `if`(irem(t, k)=0, b(n, s minus {j}, t)
          +x^k, 0)))(w*n+j), j=s)))(1+n-nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, {$0..n-1}, 0)):
seq(T(n), n=1..14);

Mathematica

b[n_, s_, w_] := b[n, s, w] = If[s == {}, 0, With[{k = 1+n-Length[s]}, Sum[With[{t = w*n + j}, If[t == 0, x, If[Mod[t, k] == 0, b[n, s ~Complement~ {j}, t] + x^k, 0]]], {j, s}]]];
T[n_] := PadRight[CoefficientList[b[n, Range[0, n-1], 0]/x, x], n];
Array[T, 14] // Flatten (* Jean-François Alcover, Feb 11 2021, after Alois P. Heinz *)

Crossrefs

Columns k=1-4 give: A000027, A334320, A333405, A333469.

Row sums give A334319.

Bisection of main diagonal (even part) gives A181736.

Cf. A111456.

Sequence in context: A359368 A353490 A226131 * A199056 A350004 A144966

Adjacent sequences: A334315 A334316 A334317 * A334319 A334320 A334321

Keyword

nonn,tabl

Author

Alois P. Heinz, Apr 22 2020

Status

approved