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.

1, 1, 2, 6, 15, 45, 136, 430, 1415, 4844, 17224, 63397, 241968, 953213, 3879822, 16250333, 70050877, 309714232, 1404000641, 6506809837, 30813282963, 148741986670, 731495853897, 3657808596354, 18588011870288, 95841754173073, 501169433939670, 2654344778727646

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)^A007580(j).

Maple

g:= proc(n) option remember; `if`(n<4, [1, 1, 2, 4][n+1],
      ((40*n^3+1084*n^2+8684*n+18480)*g(n-1) +16*(n-1)*
      (5*n^3+107*n^2+610*n+600)*g(n-2) -1024*(n-1)*(n-2)*
      (n+6)*g(n-3) -1024*(n-1)*(n-2)*(n-3)*(n+4)*g(n-4))
       /((n+7)*(n+12)*(n+15)*(n+16)))
    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, 8];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 06 2018, using code from A293112 *)

Crossrefs

Column k=8 of A293112.

Cf. A007580, A293738.

Sequence in context: A293744 A293745 A293746 * A293748 A293749 A293114

Adjacent sequences: A293744 A293745 A293746 * A293748 A293749 A293750

Keyword

nonn

Author

Alois P. Heinz, Oct 15 2017

Status

approved