cp's OEIS Frontend

A293114 Number of sets of nonempty words with a total of n letters over n-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.

%I A293114 #26 Jun 16 2018 19:33:03
%S A293114 1,1,2,6,15,45,136,430,1415,4845,17235,63509,242854,959904,3926209,
%T A293114 16564083,72097127,322898943,1487602607,7034420691,34122991199,
%U A293114 169499127425,861596397518,4475340840980,23738200183570,128427236055296,708248486616539,3977551340260517
%N A293114 Number of sets of nonempty words with a total of n letters over n-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.
%H A293114 Alois P. Heinz, <a href="/A293114/b293114.txt">Table of n, a(n) for n = 0..800</a>
%F A293114 G.f.: Product_{j>=1} (1+x^j)^A000085(j).
%F A293114 Weigh transform of A000085.
%e A293114 a(0) = 1: {}.
%e A293114 a(1) = 1: {a}.
%e A293114 a(2) = 2: {aa}, {ab}.
%e A293114 a(3) = 6: {a,aa}, {a,ab}, {aaa}, {aab}, {aba}, {abc}.
%p A293114 g:= proc(n) option remember;
%p A293114       `if`(n<2, 1, g(n-1)+(n-1)*g(n-2))
%p A293114     end:
%p A293114 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A293114       add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))
%p A293114     end:
%p A293114 a:= n-> b(n$2):
%p A293114 seq(a(n), n=0..35);
%t A293114 g[n_] := g[n] = If[n < 2, 1, g[n - 1] + (n - 1)*g[n - 2]];
%t A293114 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]* Binomial[g[i], j], {j, 0, n/i}]]];
%t A293114 a[n_] := b[n, n];
%t A293114 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 06 2018, from Maple *)
%Y A293114 Main diagonal of A293112.
%Y A293114 Row sums of A293113 and of A293815.
%Y A293114 Cf. A000085, A182172, A293110, A306009.
%K A293114 nonn
%O A293114 0,3
%A A293114 _Alois P. Heinz_, Sep 30 2017