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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.

%I A334318 #27 Feb 11 2021 14:33:15
%S A334318 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,
%T A334318 50,58,52,21,8,3,9,28,67,98,101,57,30,7,0,10,41,115,168,220,88,51,9,4,
%U A334318 1,11,45,134,275,398,315,220,126,32,10,0,12,61,206,428,690,568,503,158,32,5,1,0
%N 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.
%H A334318 Alois P. Heinz, <a href="/A334318/b334318.txt">Rows n = 1..25, flattened</a>
%e A334318 T(4,3) = 5: 102, 120, 201, 123, 321 (written in base 4):
%e A334318 T(7,2) = 15: 13, 15, 20, 24, 26, 31, 35, 40, 42, 46, 51, 53, 60, 62, 64 (written in base 7)
%e A334318 T(10,1) = 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
%e A334318 T(10,10) = 1: 3816547290.
%e A334318 Triangle T(n,k) begins:
%e A334318    1;
%e A334318    2,  1;
%e A334318    3,  1,   0;
%e A334318    4,  5,   5,   2;
%e A334318    5,  6,   6,   1,   0;
%e A334318    6, 13,  18,   8,   7,   2;
%e A334318    7, 15,  33,  34,  16,   7,   0;
%e A334318    8, 25,  50,  58,  52,  21,   8,   3;
%e A334318    9, 28,  67,  98, 101,  57,  30,   7,  0;
%e A334318   10, 41, 115, 168, 220,  88,  51,   9,  4,  1;
%e A334318   11, 45, 134, 275, 398, 315, 220, 126, 32, 10, 0;
%e A334318   12, 61, 206, 428, 690, 568, 503, 158, 32,  5, 1, 0;
%e A334318   ...
%p A334318 b:= proc(n, s, w) option remember; `if`(s={}, 0, (k-> add((t->
%p A334318       `if`(t=0, x, `if`(irem(t, k)=0, b(n, s minus {j}, t)
%p A334318           +x^k, 0)))(w*n+j), j=s)))(1+n-nops(s))
%p A334318     end:
%p A334318 T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, {$0..n-1}, 0)):
%p A334318 seq(T(n), n=1..14);
%t A334318 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 A334318 T[n_] := PadRight[CoefficientList[b[n, Range[0, n-1], 0]/x, x], n];
%t A334318 Array[T, 14] // Flatten (* _Jean-François Alcover_, Feb 11 2021, after _Alois P. Heinz_ *)
%Y A334318 Columns k=1-4 give: A000027, A334320, A333405, A333469.
%Y A334318 Row sums give A334319.
%Y A334318 Bisection of main diagonal (even part) gives A181736.
%Y A334318 Cf. A111456.
%K A334318 nonn,tabl
%O A334318 1,2
%A A334318 _Alois P. Heinz_, Apr 22 2020