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%I #25 Sep 20 2018 12:01:50
%S 1,1,0,1,1,0,1,1,2,0,1,1,4,3,0,1,1,4,8,5,0,1,1,4,14,25,7,0,1,1,4,14,
%T 43,53,11,0,1,1,4,14,67,139,148,15,0,1,1,4,14,67,223,495,328,22,0,1,1,
%U 4,14,67,343,951,1544,858,30,0,1,1,4,14,67,343,1431,3680,5111,1938,42,0
%N Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet such that within each 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="/A292712/b292712.txt">Antidiagonals n = 0..140, flattened</a>
%F G.f. of column k: Product_{j>=1} 1/(1-x^j)^A226873(j,k).
%F A(n,n) = A(n,k) for all k >= n.
%F A(n,k) = Sum_{j=0..n} A319495(n,j).
%e A(2,3) = 4: {aa}, {ab}, {ba}, {a,a}.
%e A(3,2) = 8: {aaa}, {aab}, {aba}, {baa}, {aa,a}, {ab,a}, {ba,a}, {a,a,a}.
%e A(3,3) = 14: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}, {a,a,a}.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 2, 4, 4, 4, 4, 4, 4, 4, ...
%e 0, 3, 8, 14, 14, 14, 14, 14, 14, ...
%e 0, 5, 25, 43, 67, 67, 67, 67, 67, ...
%e 0, 7, 53, 139, 223, 343, 343, 343, 343, ...
%e 0, 11, 148, 495, 951, 1431, 2151, 2151, 2151, ...
%e 0, 15, 328, 1544, 3680, 6620, 9860, 14900, 14900, ...
%e 0, 22, 858, 5111, 16239, 31539, 53739, 78939, 119259, ...
%p b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
%p add(b(n-j, j, t-1)/j!, j=i..n/t))
%p end:
%p g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
%p A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
%p g(d, k), 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 b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
%t g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]];
%t A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*g[d, k], {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 07 2018, from Maple *)
%Y Columns k=0-10 give: A000007, A000041, A292548, A292718, A292719, A292720, A292721, A292722, A292723, A292724, A292725.
%Y Rows n=0-1 give: A000012, A057427.
%Y Main diagonal gives A292713.
%Y Cf. A226873, A292795, A319495.
%K nonn,tabl
%O 0,9
%A _Alois P. Heinz_, Sep 21 2017
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