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A293747 Number of sets of nonempty words with a total of n letters over octonary 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 A293747 #9 Jun 06 2018 11:36:38
%S A293747 1,1,2,6,15,45,136,430,1415,4844,17224,63397,241968,953213,3879822,
%T A293747 16250333,70050877,309714232,1404000641,6506809837,30813282963,
%U A293747 148741986670,731495853897,3657808596354,18588011870288,95841754173073,501169433939670,2654344778727646
%N A293747 Number of sets of nonempty words with a total of n letters over octonary 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 A293747 Alois P. Heinz, <a href="/A293747/b293747.txt">Table of n, a(n) for n = 0..1000</a>
%F A293747 G.f.: Product_{j>=1} (1+x^j)^A007580(j).
%p A293747 g:= proc(n) option remember; `if`(n<4, [1, 1, 2, 4][n+1],
%p A293747       ((40*n^3+1084*n^2+8684*n+18480)*g(n-1) +16*(n-1)*
%p A293747       (5*n^3+107*n^2+610*n+600)*g(n-2) -1024*(n-1)*(n-2)*
%p A293747       (n+6)*g(n-3) -1024*(n-1)*(n-2)*(n-3)*(n+4)*g(n-4))
%p A293747        /((n+7)*(n+12)*(n+15)*(n+16)))
%p A293747     end:
%p A293747 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A293747       add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))
%p A293747     end:
%p A293747 a:= n-> b(n$2):
%p A293747 seq(a(n), n=0..35);
%t A293747 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 A293747 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 A293747 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 A293747 a[n_] := b[n, n, 8];
%t A293747 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 06 2018, using code from A293112 *)
%Y A293747 Column k=8 of A293112.
%Y A293747 Cf. A007580, A293738.
%K A293747 nonn
%O A293747 0,3
%A A293747 _Alois P. Heinz_, Oct 15 2017