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.

1, 1, 2, 6, 15, 45, 136, 430, 1414, 4835, 17143, 62843, 238245, 930418, 3741710, 15445815, 65384356, 283113205, 1252393193, 5648731817, 25945636702, 121172059749, 574764521186, 2765620022767, 13486540312370, 66587056756662, 332594340605540, 1679325348600290

Offset: 0

Author

Alois P. Heinz, Oct 15 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=7 of A293112.

Cf. A007578, A293737.

Programs

Maple

g:= proc(n) option remember; `if`(n<4, [1, 1, 2, 4][n+1],
      ((4*n^3+78*n^2+424*n+495)*g(n-1) +(n-1)*(34*n^2+280*n+
       305)*g(n-2) -2*(n-1)*(n-2)*(38*n+145)*g(n-3) -105*(n-1)
       *(n-2)*(n-3)*g(n-4))/((n+6)*(n+10)*(n+12)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);

Mathematica

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]];
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]]]]]];
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}]]];
a[n_] := b[n, n, 7];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 06 2018, using code from A293112 *)

Formula

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