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A319518 Number of sets of nonempty words with a total of n letters over n-ary alphabet such that if a letter occurs in the set all predecessors occur at least once.

%I A319518 #17 Mar 10 2022 09:31:26
%S A319518 1,1,4,27,218,2178,25529,343392,5205948,87740878,1626182463,
%T A319518 32852520594,718169744206,16883948532684,424649281630018,
%U A319518 11374387591643065,323183885622356184,9706973096869527210,307248234238900686688,10220414166250239718518
%N A319518 Number of sets of nonempty words with a total of n letters over n-ary alphabet such that if a letter occurs in the set all predecessors occur at least once.
%H A319518 Alois P. Heinz, <a href="/A319518/b319518.txt">Table of n, a(n) for n = 0..300</a>
%e A319518 a(0) = 1: {}.
%e A319518 a(1) = 1: {a}.
%e A319518 a(2) = 4: {aa}, {ab}, {ba}, {a,b}.
%e A319518 a(3) = 27: {aaa}, {aab}, {aba}, {abb}, {abc}, {acb}, {baa}, {bab}, {bac}, {bba}, {bca}, {cab}, {cba}, {a,aa}, {a,ab}, {a,ba}, {a,bb}, {a,bc}, {a,cb}, {aa,b}, {ab,b}, {ab,c}, {ac,b}, {b,ba}, {b,ca}, {ba,c}, {a,b,c}.
%p A319518 h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A319518       add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
%p A319518     end:
%p A319518 a:= n-> add(add((-1)^i*binomial(k, i)*h(n$2, k-i), i=0..k), k=0..n):
%p A319518 seq(a(n), n=0..20);
%t A319518 h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0,
%t A319518      Sum[h[n - i*j, i - 1, k]*Binomial[k^i, j], {j, 0, n/i}]]];
%t A319518 a[n_] := Sum[Sum[(-1)^i*Binomial[k, i]*h[n, n, k-i], {i, 0, k}], {k, 0, n}];
%t A319518 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 10 2022, after _Alois P. Heinz_ *)
%Y A319518 Row sums of A319501.
%Y A319518 Cf. A257741.
%K A319518 nonn
%O A319518 0,3
%A A319518 _Alois P. Heinz_, Sep 21 2018