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A293808 Number T(n,k) of multisets of exactly k nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A293808 #22 Aug 12 2020 18:54:36
%S A293808 1,0,1,0,2,1,0,4,2,1,0,10,7,2,1,0,26,18,7,2,1,0,76,56,22,7,2,1,0,232,
%T A293808 168,68,22,7,2,1,0,764,543,218,73,22,7,2,1,0,2620,1792,721,234,73,22,
%U A293808 7,2,1,0,9496,6187,2438,791,240,73,22,7,2,1,0,35696,22088,8491,2702,811,240,73,22,7,2,1
%N A293808 Number T(n,k) of multisets of exactly k nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A293808 Alois P. Heinz, <a href="/A293808/b293808.txt">Rows n = 0..200, flattened</a>
%H A293808 <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>
%F A293808 G.f.: Product_{j>=1} 1/(1-y*x^j)^A000085(j).
%e A293808 T(0,0) = 1: {}.
%e A293808 T(3,1) = 4: {aaa}, {aab}, {aba}, {abc}.
%e A293808 T(3,2) = 2: {a,aa}, {a,ab}.
%e A293808 T(3,3) = 1: {a,a,a}.
%e A293808 T(4,2) = 7: {a,aaa}, {a,aab}, {a,aba}, {a,abc}, {aa,aa}, {aa,ab}, {ab,ab}.
%e A293808 Triangle T(n,k) begins:
%e A293808   1;
%e A293808   0,    1;
%e A293808   0,    2,    1;
%e A293808   0,    4,    2,    1;
%e A293808   0,   10,    7,    2,   1;
%e A293808   0,   26,   18,    7,   2,   1;
%e A293808   0,   76,   56,   22,   7,   2,  1;
%e A293808   0,  232,  168,   68,  22,   7,  2,  1;
%e A293808   0,  764,  543,  218,  73,  22,  7,  2, 1;
%e A293808   0, 2620, 1792,  721, 234,  73, 22,  7, 2, 1;
%e A293808   0, 9496, 6187, 2438, 791, 240, 73, 22, 7, 2, 1;
%e A293808   ...
%p A293808 g:= proc(n) option remember; `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
%p A293808 b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, x^n,
%p A293808       add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)))
%p A293808     end:
%p A293808 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
%p A293808 seq(T(n), n=0..15);
%t A293808 g[n_] := g[n] = If[n < 2, 1, g[n - 1] + (n - 1)*g[n - 2]] ;
%t A293808 b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, x^n, Sum[Binomial[g[i] + j - 1, j]*b[n - i*j, i - 1]*x^j, {j, 0, n/i}]]];
%t A293808 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
%t A293808 Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Jun 04 2018, from Maple *)
%Y A293808 Columns k=0-10 give: A000007, A000085 (for n>0), A294004, A294005, A294006, A294007, A294008, A294009, A294010, A294011, A294012.
%Y A293808 Row sums give: A293110.
%Y A293808 T(2n,n) gives A293809.
%Y A293808 Cf. A293815.
%K A293808 nonn,tabl
%O A293808 0,5
%A A293808 _Alois P. Heinz_, Oct 16 2017