A293744 Number of sets of nonempty words with a total of n letters over quinary 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, 135, 422, 1357, 4503, 15301, 53225, 189070, 684540, 2522194, 9441960, 35867225, 138080428, 538155330, 2121211604, 8448577175, 33974559322, 137842934746, 563885092371, 2324435490519, 9650120731330, 40329864236526, 169593208033062

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)^A049401(j).

Maple

g:= proc(n) option remember;
      `if`(n<3, [1, 1, 2][n+1], ((3*n^2+17*n+15)*g(n-1)
       +(n-1)*(13*n+9)*g(n-2) -15*(n-1)*(n-2)*g(n-3)) /
       ((n+4)*(n+6)))
    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

g[n_] := g[n] = If[n<3, {1, 1, 2}[[n+1]], ((3*n^2 + 17*n + 15)*g[n-1] + (n-1)*(13*n + 9)*g[n-2] - 15*(n-1)*(n-2)*g[n-3]) / ((n+4)*(n+6))];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]* Binomial[g[i], j], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 06 2018, from Maple *)

Crossrefs

Column k=5 of A293112.

Cf. A049401, A293735.

Sequence in context: A293743 A360274 A001444 * A293745 A293746 A293747

Adjacent sequences: A293741 A293742 A293743 * A293745 A293746 A293747

Keyword

nonn

Author

Alois P. Heinz, Oct 15 2017

Status

approved