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A327545 Triangle T(n,k) read by rows giving the number of zeroless polydivisible numbers in base n that have k distinct digits with 1 <= k <= n-1.

%I A327545 #35 Sep 18 2019 10:32:59
%S A327545 1,4,0,5,2,2,10,14,8,0,7,14,20,2,2,26,39,84,60,27,0,11,47,108,95,63,3,
%T A327545 3,20,101,233,369,289,79,17,0,19,86,306,475,714,409,146,1,1,32,201,
%U A327545 979,2048,3581,3474,1925,449,51,0,17,114,507,1273,2224,2239,1074,230,35,0,0
%N A327545 Triangle T(n,k) read by rows giving the number of zeroless polydivisible numbers in base n that have k distinct digits with 1 <= k <= n-1.
%C A327545 For k >= n there is no k-digit zeroless polydivisible number in base n.
%H A327545 Seiichi Manyama, <a href="/A327545/b327545.txt">Rows n = 2..18, flattened</a>
%H A327545 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polydivisible_number">Polydivisible number</a>.
%e A327545 n | zeroless polydivisible numbers in base n
%e A327545 --+------------------------------------------
%e A327545 2 | [1]
%e A327545 3 | [1, 2, 11, 22]
%e A327545 4 | [1, 2, 3, 22, 222],  [12, 32], [123, 321]
%e A327545 So T(2,1) = 1, T(3,1) = 4, T(3,2) = 0, T(4,1) = 5, T(4,2) = 2, T(4,3) = 2.
%e A327545 Triangle begins:
%e A327545 n\k  |  1    2    3    4    5    6    7  8  9
%e A327545 -----+----------------------------------------
%e A327545    2 |  1;
%e A327545    3 |  4,   0;
%e A327545    4 |  5,   2,   2;
%e A327545    5 | 10,  14,   8,   0;
%e A327545    6 |  7,  14,  20,   2,   2;
%e A327545    7 | 26,  39,  84,  60,  27,   0;
%e A327545    8 | 11,  47, 108,  95,  63,   3,   3;
%e A327545    9 | 20, 101, 233, 369, 289,  79,  17, 0;
%e A327545   10 | 19,  86, 306, 475, 714, 409, 146, 1, 1;
%o A327545 (Ruby)
%o A327545 def A(n)
%o A327545   d = 0
%o A327545   a = (1..n - 1).map{|i| [i]}
%o A327545   ary = [n - 1] + Array.new(n - 2, 0)
%o A327545   while d < n - 2
%o A327545     d += 1
%o A327545     b = []
%o A327545     a.each{|i|
%o A327545       (1..n - 1).each{|j|
%o A327545         m = i.clone + [j]
%o A327545         if (0..d).inject(0){|s, k| s + m[k] * n ** (d - k)} % (d + 1) == 0
%o A327545           b << m
%o A327545           ary[m.uniq.size - 1] += 1
%o A327545         end
%o A327545       }
%o A327545     }
%o A327545     a = b
%o A327545   end
%o A327545   ary
%o A327545 end
%o A327545 def A327545(n)
%o A327545   (2..n).map{|i| A(i)}.flatten
%o A327545 end
%o A327545 p A327545(10)
%Y A327545 Row sums give A324020.
%Y A327545 T(2*n,2*n-1) gives A181736.
%Y A327545 T(n,1) gives A327577.
%Y A327545 Cf. A324019, A324205.
%K A327545 nonn,base,tabl
%O A327545 2,2
%A A327545 _Seiichi Manyama_, Sep 16 2019