A319501 Number T(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet such that all k letters occur at least once in the set; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 3, 0, 2, 12, 13, 0, 2, 38, 105, 73, 0, 3, 110, 588, 976, 501, 0, 4, 302, 2811, 8416, 9945, 4051, 0, 5, 806, 12354, 59488, 121710, 111396, 37633, 0, 6, 2109, 51543, 375698, 1185360, 1830822, 1366057, 394353, 0, 8, 5450, 207846, 2209276, 10096795, 23420022, 28969248, 18235680, 4596553

Offset

0,6

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A292804(n,k-i).

Example

T(2,2) = 3: {ab}, {ba}, {a,b}.
T(3,2) = 12: {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {a,ab}, {a,ba}, {a,bb}, {aa,b}, {ab,b}, {b,ba}.
T(4,2) = 38: {aaab}, {aaba}, {aabb}, {abaa}, {abab}, {abba}, {abbb}, {baaa}, {baab}, {baba}, {babb}, {bbaa}, {bbab}, {bbba}, {a,aab}, {a,aba}, {a,abb}, {a,baa}, {a,bab}, {a,bba}, {a,bbb}, {aa,ab}, {aa,ba}, {aa,bb}, {aaa,b}, {aab,b}, {ab,ba}, {ab,bb}, {aba,b}, {abb,b}, {b,baa}, {b,bab}, {b,bba}, {ba,bb}, {a,aa,b}, {a,ab,b}, {a,b,ba}, {a,b,bb}.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,    3;
  0, 2,   12,    13;
  0, 2,   38,   105,     73;
  0, 3,  110,   588,    976,     501;
  0, 4,  302,  2811,   8416,    9945,    4051;
  0, 5,  806, 12354,  59488,  121710,  111396,   37633;
  0, 6, 2109, 51543, 375698, 1185360, 1830822, 1366057, 394353;

Maple

h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
    end:
T:= (n, k)-> add((-1)^i*binomial(k, i)*h(n$2, k-i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

h[n_, i_, k_] := h[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[h[n-i*j, i-1, k]* Binomial[k^i, j], {j, 0, n/i}]]];
T[n_, k_] := Sum[(-1)^i Binomial[k, i] h[n, n, k-i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 05 2020, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A000007, A000009 (for n>0), A320203, A320204, A320205, A320206, A320207, A320208, A320209, A320210, A320211.

Main diagonal gives A000262.

Row sums give A319518.

T(2n,n) gives A319519.

Cf. A257740, A292804.

Sequence in context: A326602 A256548 A239098 * A302224 A302670 A302472

Adjacent sequences: A319498 A319499 A319500 * A319502 A319503 A319504

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 20 2018

Status

approved