cp's OEIS Frontend

A293746 Number of sets of nonempty words with a total of n letters over septenary 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 A293746 #10 Jun 06 2018 11:36:31
%S A293746 1,1,2,6,15,45,136,430,1414,4835,17143,62843,238245,930418,3741710,
%T A293746 15445815,65384356,283113205,1252393193,5648731817,25945636702,
%U A293746 121172059749,574764521186,2765620022767,13486540312370,66587056756662,332594340605540,1679325348600290
%N A293746 Number of sets of nonempty words with a total of n letters over septenary 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 A293746 Alois P. Heinz, <a href="/A293746/b293746.txt">Table of n, a(n) for n = 0..1000</a>
%F A293746 G.f.: Product_{j>=1} (1+x^j)^A007578(j).
%p A293746 g:= proc(n) option remember; `if`(n<4, [1, 1, 2, 4][n+1],
%p A293746       ((4*n^3+78*n^2+424*n+495)*g(n-1) +(n-1)*(34*n^2+280*n+
%p A293746        305)*g(n-2) -2*(n-1)*(n-2)*(38*n+145)*g(n-3) -105*(n-1)
%p A293746        *(n-2)*(n-3)*g(n-4))/((n+6)*(n+10)*(n+12)))
%p A293746     end:
%p A293746 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A293746       add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))
%p A293746     end:
%p A293746 a:= n-> b(n$2):
%p A293746 seq(a(n), n=0..35);
%t A293746 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 A293746 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 A293746 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 A293746 a[n_] := b[n, n, 7];
%t A293746 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 06 2018, using code from A293112 *)
%Y A293746 Column k=7 of A293112.
%Y A293746 Cf. A007578, A293737.
%K A293746 nonn
%O A293746 0,3
%A A293746 _Alois P. Heinz_, Oct 15 2017