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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-François 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.)