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%I #23 Jan 05 2020 05:36:05
%S 1,0,1,0,1,3,0,2,12,13,0,2,38,105,73,0,3,110,588,976,501,0,4,302,2811,
%T 8416,9945,4051,0,5,806,12354,59488,121710,111396,37633,0,6,2109,
%U 51543,375698,1185360,1830822,1366057,394353,0,8,5450,207846,2209276,10096795,23420022,28969248,18235680,4596553
%N Number T(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet such that all k letters occur at least once in the set; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A319501/b319501.txt">Rows n = 0..140, flattened</a>
%F T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A292804(n,k-i).
%e T(2,2) = 3: {ab}, {ba}, {a,b}.
%e T(3,2) = 12: {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {a,ab}, {a,ba}, {a,bb}, {aa,b}, {ab,b}, {b,ba}.
%e T(4,2) = 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}.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 3;
%e 0, 2, 12, 13;
%e 0, 2, 38, 105, 73;
%e 0, 3, 110, 588, 976, 501;
%e 0, 4, 302, 2811, 8416, 9945, 4051;
%e 0, 5, 806, 12354, 59488, 121710, 111396, 37633;
%e 0, 6, 2109, 51543, 375698, 1185360, 1830822, 1366057, 394353;
%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 T:= (n, k)-> add((-1)^i*binomial(k, i)*h(n$2, k-i), i=0..k):
%p seq(seq(T(n, k), k=0..n), n=0..12);
%t 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}]]];
%t T[n_, k_] := Sum[(-1)^i Binomial[k, i] h[n, n, k-i], {i, 0, k}];
%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 05 2020, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000007, A000009 (for n>0), A320203, A320204, A320205, A320206, A320207, A320208, A320209, A320210, A320211.
%Y Main diagonal gives A000262.
%Y Row sums give A319518.
%Y T(2n,n) gives A319519.
%Y Cf. A257740, A292804.
%K nonn,tabl
%O 0,6
%A _Alois P. Heinz_, Sep 20 2018
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