%I #17 Mar 10 2022 09:31:26

%S 1,1,4,27,218,2178,25529,343392,5205948,87740878,1626182463,

%T 32852520594,718169744206,16883948532684,424649281630018,

%U 11374387591643065,323183885622356184,9706973096869527210,307248234238900686688,10220414166250239718518

%N 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 Alois P. Heinz, <a href="/A319518/b319518.txt">Table of n, a(n) for n = 0..300</a>

%e a(0) = 1: {}.

%e a(1) = 1: {a}.

%e a(2) = 4: {aa}, {ab}, {ba}, {a,b}.

%e 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 h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))

%p end:

%p a:= n-> add(add((-1)^i*binomial(k, i)*h(n$2, k-i), i=0..k), k=0..n):

%p seq(a(n), n=0..20);

%t h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0,

%t Sum[h[n - i*j, i - 1, k]*Binomial[k^i, j], {j, 0, n/i}]]];

%t a[n_] := Sum[Sum[(-1)^i*Binomial[k, i]*h[n, n, k-i], {i, 0, k}], {k, 0, n}];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 10 2022, after _Alois P. Heinz_ *)

%Y Row sums of A319501.

%Y Cf. A257741.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 21 2018