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A293108 Number A(n,k) of multisets of nonempty words with a total of n letters over k-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; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A293108 #26 Sep 20 2018 12:12:20
%S A293108 1,1,0,1,1,0,1,1,2,0,1,1,3,3,0,1,1,3,6,5,0,1,1,3,7,15,7,0,1,1,3,7,19,
%T A293108 31,11,0,1,1,3,7,20,48,73,15,0,1,1,3,7,20,53,131,155,22,0,1,1,3,7,20,
%U A293108 54,157,348,351,30,0,1,1,3,7,20,54,163,455,954,755,42,0
%N A293108 Number A(n,k) of multisets of nonempty words with a total of n letters over k-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; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A293108 Alois P. Heinz, <a href="/A293108/b293108.txt">Antidiagonals n = 0..80, flattened</a>
%F A293108 G.f. of column k: Product_{j>=1} 1/(1-x^j)^A182172(j,k).
%F A293108 A(n,k) = Sum_{j=0..k} A293109(n,j).
%F A293108 A(n,n) = A(n,k) for all k >= n.
%e A293108 Square array A(n,k) begins:
%e A293108   1,  1,   1,   1,   1,   1,   1,   1, ...
%e A293108   0,  1,   1,   1,   1,   1,   1,   1, ...
%e A293108   0,  2,   3,   3,   3,   3,   3,   3, ...
%e A293108   0,  3,   6,   7,   7,   7,   7,   7, ...
%e A293108   0,  5,  15,  19,  20,  20,  20,  20, ...
%e A293108   0,  7,  31,  48,  53,  54,  54,  54, ...
%e A293108   0, 11,  73, 131, 157, 163, 164, 164, ...
%e A293108   0, 15, 155, 348, 455, 492, 499, 500, ...
%p A293108 h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]
%p A293108     <j, 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
%p A293108 g:= proc(n, i, l) option remember;
%p A293108       `if`(n=0, h(l), `if`(i<1, 0, `if`(i=1, h([l[], 1$n]),
%p A293108         g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
%p A293108     end:
%p A293108 A:= proc(n, k) option remember; `if`(n=0, 1, add(add(g(d, k, [])
%p A293108       *d, d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)
%p A293108     end:
%p A293108 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A293108 h[l_] := Function [n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[ l[[k]] < j, 0, 1], {k, i+1, n}], {j, 1, l[[i]]}], {i, n}]][Length[l]];
%t A293108 g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Table[1, n]]], g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Append[l, i]]]]]];
%t A293108 A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[g[d, k, {}]*d, {d, Divisors[j] }]*A[n - j, k], {j, 1, n}]/n];
%t A293108 Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Jun 03 2018, from Maple *)
%Y A293108 Columns k=0-10 give: A000007, A000041, A293732, A293733, A293734, A293735, A293736, A293737, A293738, A293739, A293740.
%Y A293108 Main diagonal gives A293110.
%Y A293108 Cf. A182172, A293109, A293112.
%K A293108 nonn,tabl
%O A293108 0,9
%A A293108 _Alois P. Heinz_, Sep 30 2017