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.

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

Author

Alois P. Heinz, Sep 23 2017

Keywords

• nonn
• tabl

Examples

			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, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened

Crossrefs

Columns k=0-10 give: A000007, A000009, A102866, A256142, A292838, A292839, A292840, A292841, A292842, A292843, A292844.

Rows n=0-2 give: A000012, A001477, A000326.

Main diagonal gives A292805.

Cf. A144074, A292795, A319501.

Programs

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 *)

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).