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A293114 Number of sets of nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter. 6
1, 1, 2, 6, 15, 45, 136, 430, 1415, 4845, 17235, 63509, 242854, 959904, 3926209, 16564083, 72097127, 322898943, 1487602607, 7034420691, 34122991199, 169499127425, 861596397518, 4475340840980, 23738200183570, 128427236055296, 708248486616539, 3977551340260517 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
FORMULA
G.f.: Product_{j>=1} (1+x^j)^A000085(j).
Weigh transform of A000085.
EXAMPLE
a(0) = 1: {}.
a(1) = 1: {a}.
a(2) = 2: {aa}, {ab}.
a(3) = 6: {a,aa}, {a,ab}, {aaa}, {aab}, {aba}, {abc}.
MAPLE
g:= proc(n) option remember;
`if`(n<2, 1, g(n-1)+(n-1)*g(n-2))
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 < 2, 1, g[n - 1] + (n - 1)*g[n - 2]];
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
Main diagonal of A293112.
Row sums of A293113 and of A293815.
Sequence in context: A293747 A293748 A293749 * A322188 A317976 A045628
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 30 2017
STATUS
approved

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)