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A293745 Number of sets of nonempty words with a total of n letters over senary 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 A293745 #10 Jun 06 2018 11:36:25
%S A293745 1,1,2,6,15,45,136,429,1406,4771,16749,60453,224948,857010,3350574,
%T A293745 13366375,54494538,226020624,954737292,4092229831,17813005015,
%U A293745 78509835288,350592604663,1582430253294,7223028969003,33275812688050,154790795962448,725871751770492
%N A293745 Number of sets of nonempty words with a total of n letters over senary 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 A293745 Alois P. Heinz, <a href="/A293745/b293745.txt">Table of n, a(n) for n = 0..1000</a>
%F A293745 G.f.: Product_{j>=1} (1+x^j)^A007579(j).
%p A293745 g:= proc(n) option remember;
%p A293745       `if`(n<4, [1, 1, 2, 4][n+1], ((20*n^2+184*n+336)*g(n-1)
%p A293745        +4*(n-1)*(10*n^2+58*n+33)*g(n-2) -144*(n-1)*(n-2)*g(n-3)
%p A293745        -144*(n-1)*(n-2)*(n-3)*g(n-4))/ ((n+5)*(n+8)*(n+9)))
%p A293745     end:
%p A293745 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A293745       add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))
%p A293745     end:
%p A293745 a:= n-> b(n$2):
%p A293745 seq(a(n), n=0..35);
%t A293745 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, 1, n}]][Length[l]];
%t A293745 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 A293745 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*Binomial[g[i, k, {}], j], {j, 0, n/i}]]];
%t A293745 a[n_] := b[n, n, 6];
%t A293745 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 06 2018, using code from A293112 *)
%Y A293745 Column k=6 of A293112.
%Y A293745 Cf. A007579, A293736.
%K A293745 nonn
%O A293745 0,3
%A A293745 _Alois P. Heinz_, Oct 15 2017