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

0,3

Links

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

Formula

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

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 *)

Crossrefs

Column k=7 of A293112.

Cf. A007578, A293737.

Sequence in context: A001444 A293744 A293745 * A293747 A293748 A293749

Adjacent sequences: A293743 A293744 A293745 * A293747 A293748 A293749

Keyword

nonn

Author

Alois P. Heinz, Oct 15 2017

Status

approved