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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

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

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-Fran├žois Alcover, Jun 03 2018, from Maple *)

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.

Sequence in context: A286933 A295860 A118345 * A118350 A183135 A294042

Adjacent sequences:  A292801 A292802 A292803 * A292805 A292806 A292807

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 23 2017

STATUS

approved

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Last modified April 5 22:56 EDT 2020. Contains 333260 sequences. (Running on oeis4.)