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A292804
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.
15
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, 132, 225, 116, 4, 0, 1, 7, 51, 260, 729, 927, 310, 5, 0, 1, 8, 70, 452, 1805, 4000, 3729, 816, 6, 0, 1, 9, 92, 721, 3777, 12376, 21488, 14787, 2121, 8, 0
OFFSET
0,8
LINKS
FORMULA
G.f. of column k: Product_{j>=1} (1+x^j)^(k^j).
A(n,k) = Sum_{i=0..k} C(k,i) * A319501(n,i).
EXAMPLE
A(2,2) = 5: {aa}, {ab}, {ba}, {bb}, {a,b}.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, ...
0, 1, 5, 12, 22, 35, 51, 70, ...
0, 2, 16, 55, 132, 260, 452, 721, ...
0, 2, 42, 225, 729, 1805, 3777, 7042, ...
0, 3, 116, 927, 4000, 12376, 31074, 67592, ...
0, 4, 310, 3729, 21488, 83175, 250735, 636517, ...
0, 5, 816, 14787, 113760, 550775, 1993176, 5904746, ...
MAPLE
h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
end:
A:= (n, k)-> h(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
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}]]];
A[n_, k_] := h[n, n, k];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)
CROSSREFS
Rows n=0-2 give: A000012, A001477, A000326.
Main diagonal gives A292805.
Sequence in context: A286933 A295860 A118345 * A118350 A361950 A183135
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 23 2017
STATUS
approved