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%I #27 Sep 21 2018 17:25:59
%S 1,1,0,1,1,0,1,2,1,0,1,3,5,2,0,1,4,12,16,2,0,1,5,22,55,42,3,0,1,6,35,
%T 132,225,116,4,0,1,7,51,260,729,927,310,5,0,1,8,70,452,1805,4000,3729,
%U 816,6,0,1,9,92,721,3777,12376,21488,14787,2121,8,0
%N Number A(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H Alois P. Heinz, <a href="/A292804/b292804.txt">Antidiagonals n = 0..140, flattened</a>
%F G.f. of column k: Product_{j>=1} (1+x^j)^(k^j).
%F A(n,k) = Sum_{i=0..k} C(k,i) * A319501(n,i).
%e A(2,2) = 5: {aa}, {ab}, {ba}, {bb}, {a,b}.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, 6, 7, ...
%e 0, 1, 5, 12, 22, 35, 51, 70, ...
%e 0, 2, 16, 55, 132, 260, 452, 721, ...
%e 0, 2, 42, 225, 729, 1805, 3777, 7042, ...
%e 0, 3, 116, 927, 4000, 12376, 31074, 67592, ...
%e 0, 4, 310, 3729, 21488, 83175, 250735, 636517, ...
%e 0, 5, 816, 14787, 113760, 550775, 1993176, 5904746, ...
%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, k)-> h(n$2, k):
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%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 A[n_, k_] := h[n, n, k];
%t Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Jun 03 2018, from Maple *)
%Y Columns k=0-10 give: A000007, A000009, A102866, A256142, A292838, A292839, A292840, A292841, A292842, A292843, A292844.
%Y Rows n=0-2 give: A000012, A001477, A000326.
%Y Main diagonal gives A292805.
%Y Cf. A144074, A292795, A319501.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Sep 23 2017
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