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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. 5
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 (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)^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

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Last modified May 7 22:52 EDT 2021. Contains 343652 sequences. (Running on oeis4.)