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