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
Examples
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; ...
Links
- Alois P. Heinz, Rows n = 1..25, flattened
Crossrefs
Programs
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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 *)