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

Author

Alois P. Heinz, Sep 21 2018

Keywords

• nonn

Examples

			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}.
		

Links

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

Crossrefs

Cf. A257742, A319501.

Programs

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 *)

Formula

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