cp's OEIS Frontend

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.

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