A319519 Number of sets of nonempty words with a total of 2n letters over n-ary alphabet such that all n letters occur at least once in the set.

1, 1, 38, 2811, 375698, 78808600, 23761098837, 9706198156760, 5148887208055692, 3435636331820210328, 2812707955072045999940, 2769473851247907714803299, 3226373218837374171864997818, 4386692184929838579321027664266, 6880627149087717821279760600127300

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..214

Formula

a(n) = A319501(2n,n).

Example

a(0) = 1: {}.
a(1) = 1: {aa}.
a(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}.

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:
a:= n-> add((-1)^i*binomial(n, i)*h(2*n$2, n-i), i=0..n):
seq(a(n), n=0..15);

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}]]];
a[n_] := Sum[(-1)^i*Binomial[n, i]*h[2n, 2n, n-i], {i, 0, n}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, May 10 2022, after Alois P. Heinz *)

Crossrefs

Cf. A257742, A319501.

Sequence in context: A272036 A055605 A217342 * A221111 A322331 A282962

Adjacent sequences: A319516 A319517 A319518 * A319520 A319521 A319522

Keyword

nonn

Author

Alois P. Heinz, Sep 21 2018

Status

approved