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A292796 Number of sets 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.

%I A292796 #16 Jan 02 2021 12:20:01
%S A292796 1,1,3,13,60,326,2065,14508,116845,1039459,10339365,112376487,
%T A292796 1339665295,17256611005,240193792120,3578746993871,56986570945387,
%U A292796 963868021665359,17281651020455445,327058650473873893,6519981694119182165,136489249161324882063
%N A292796 Number of sets 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 A292796 Alois P. Heinz, <a href="/A292796/b292796.txt">Table of n, a(n) for n = 0..450</a>
%F A292796 a(n) = [x^n] Product_{j=1..n} (1+x^j)^A226873(j,n).
%F A292796 a(n) = A292795(n,n).
%F A292796 a(n) ~ c * n!, where c = A247551 = 2.529477472079152648... - _Vaclav Kotesovec_, Sep 28 2017
%e A292796 a(0) = 1: {}.
%e A292796 a(1) = 1: {a}.
%e A292796 a(2) = 3: {aa}, {ab}, {ba}.
%e A292796 a(3) = 13: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}.
%p A292796 b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
%p A292796       add(b(n-j, j, t-1)/j!, j=i..n/t))
%p A292796     end:
%p A292796 g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
%p A292796 h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A292796       add(h(n-i*j, i-1, k)*binomial(g(i, k), j), j=0..n/i)))
%p A292796     end:
%p A292796 a:= n-> h(n$3):
%p A292796 seq(a(n), n=0..30);
%t A292796 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 A292796 g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]];
%t A292796 h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[h[n - i*j, i - 1, k]*Binomial[g[i, k], j], {j, 0, n/i}]]];
%t A292796 a[n_] := h[n, n, n];
%t A292796 a /@ Range[0, 30] (* _Jean-François Alcover_, Jan 02 2021, after _Alois P. Heinz_ *)
%Y A292796 Main diagonal of A292795.
%Y A292796 Row sums of A319498.
%Y A292796 Cf. A226873, A292713.
%K A292796 nonn
%O A292796 0,3
%A A292796 _Alois P. Heinz_, Sep 23 2017