A293746 - OEIS

%I #10 Jun 06 2018 11:36:31

%S 1,1,2,6,15,45,136,430,1414,4835,17143,62843,238245,930418,3741710,

%T 15445815,65384356,283113205,1252393193,5648731817,25945636702,

%U 121172059749,574764521186,2765620022767,13486540312370,66587056756662,332594340605540,1679325348600290

%N 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 Alois P. Heinz, <a href="/A293746/b293746.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Product_{j>=1} (1+x^j)^A007578(j).

%p g:= proc(n) option remember; `if`(n<4, [1, 1, 2, 4][n+1],

%p       ((4*n^3+78*n^2+424*n+495)*g(n-1) +(n-1)*(34*n^2+280*n+

%p        305)*g(n-2) -2*(n-1)*(n-2)*(38*n+145)*g(n-3) -105*(n-1)

%p        *(n-2)*(n-3)*g(n-4))/((n+6)*(n+10)*(n+12)))

%p     end:

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..35);

%t 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 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 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 a[n_] := b[n, n, 7];

%t Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 06 2018, using code from A293112 *)

%Y Column k=7 of A293112.

%Y Cf. A007578, A293737.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Oct 15 2017