%I #22 Dec 19 2020 02:28:51

%S 1,1,4,14,67,343,2151,14900,119259,1055520,10465854,113479756,

%T 1350508150,17373376892,241576630993,3596468789967,57232276979726,

%U 967517444008250,17339617861447844,328037083000497867,6537494747743375847,136820214583596515519

%N Number of multisets of nonempty words with a total of n letters over n-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

%H Alois P. Heinz, <a href="/A292713/b292713.txt">Table of n, a(n) for n = 0..450</a>

%F a(n) = [x^n] Product_{j=1..n} 1/(1-x^j)^A226873(j,n).

%F a(n) = A292712(n,n).

%F a(n) ~ c * n!, where c = A247551 = 2.5294774720791526... - _Vaclav Kotesovec_, Oct 05 2017

%e a(0) = 1: {}.

%e a(1) = 1: {a}.

%e a(2) = 4: {aa}, {ab}, {ba}, {a,a}.

%e a(3) = 14: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}, {a,a,a}.

%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 a:= n-> A(n$2):

%p seq(a(n), n=0..25);

%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 a[n_] := A[n, n];

%t a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 19 2020, after _Alois P. Heinz_ *)

%Y Main diagonal of A292712.

%Y Row sums of A319495.

%Y Cf. A226873, A292796.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 21 2017