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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.

%I A319519 #19 May 10 2022 07:50:36
%S A319519 1,1,38,2811,375698,78808600,23761098837,9706198156760,
%T A319519 5148887208055692,3435636331820210328,2812707955072045999940,
%U A319519 2769473851247907714803299,3226373218837374171864997818,4386692184929838579321027664266,6880627149087717821279760600127300
%N 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.
%H A319519 Alois P. Heinz, <a href="/A319519/b319519.txt">Table of n, a(n) for n = 0..214</a>
%F A319519 a(n) = A319501(2n,n).
%e A319519 a(0) = 1: {}.
%e A319519 a(1) = 1: {aa}.
%e A319519 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}.
%p A319519 h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A319519       add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
%p A319519     end:
%p A319519 a:= n-> add((-1)^i*binomial(n, i)*h(2*n$2, n-i), i=0..n):
%p A319519 seq(a(n), n=0..15);
%t A319519 h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0,
%t A319519      Sum[h[n-i*j, i-1, k]*Binomial[k^i, j], {j, 0, n/i}]]];
%t A319519 a[n_] := Sum[(-1)^i*Binomial[n, i]*h[2n, 2n, n-i], {i, 0, n}];
%t A319519 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, May 10 2022, after _Alois P. Heinz_ *)
%Y A319519 Cf. A257742, A319501.
%K A319519 nonn
%O A319519 0,3
%A A319519 _Alois P. Heinz_, Sep 21 2018