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A293745 Number of sets of nonempty words with a total of n letters over senary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter. 5
1, 1, 2, 6, 15, 45, 136, 429, 1406, 4771, 16749, 60453, 224948, 857010, 3350574, 13366375, 54494538, 226020624, 954737292, 4092229831, 17813005015, 78509835288, 350592604663, 1582430253294, 7223028969003, 33275812688050, 154790795962448, 725871751770492 (list; graph; refs; listen; history; text; internal format)
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)^A007579(j).

MAPLE

g:= proc(n) option remember;

      `if`(n<4, [1, 1, 2, 4][n+1], ((20*n^2+184*n+336)*g(n-1)

       +4*(n-1)*(10*n^2+58*n+33)*g(n-2) -144*(n-1)*(n-2)*g(n-3)

       -144*(n-1)*(n-2)*(n-3)*g(n-4))/ ((n+5)*(n+8)*(n+9)))

    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, 6];

Table[a[n], {n, 0, 35}] (* Jean-Fran├žois Alcover, Jun 06 2018, using code from A293112 *)

CROSSREFS

Column k=6 of A293112.

Cf. A007579, A293736.

Sequence in context: A293743 A001444 A293744 * A293746 A293747 A293748

Adjacent sequences:  A293742 A293743 A293744 * A293746 A293747 A293748

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Oct 15 2017

STATUS

approved

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Last modified May 6 17:00 EDT 2021. Contains 343586 sequences. (Running on oeis4.)