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A293808
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Number T(n,k) of multisets of exactly k 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; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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13
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1, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 10, 7, 2, 1, 0, 26, 18, 7, 2, 1, 0, 76, 56, 22, 7, 2, 1, 0, 232, 168, 68, 22, 7, 2, 1, 0, 764, 543, 218, 73, 22, 7, 2, 1, 0, 2620, 1792, 721, 234, 73, 22, 7, 2, 1, 0, 9496, 6187, 2438, 791, 240, 73, 22, 7, 2, 1, 0, 35696, 22088, 8491, 2702, 811, 240, 73, 22, 7, 2, 1
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OFFSET
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0,5
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LINKS
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Alois P. Heinz, Rows n = 0..200, flattened
Index entries for triangles generated by the Multiset Transformation
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FORMULA
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G.f.: Product_{j>=1} 1/(1-y*x^j)^A000085(j).
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EXAMPLE
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T(0,0) = 1: {}.
T(3,1) = 4: {aaa}, {aab}, {aba}, {abc}.
T(3,2) = 2: {a,aa}, {a,ab}.
T(3,3) = 1: {a,a,a}.
T(4,2) = 7: {a,aaa}, {a,aab}, {a,aba}, {a,abc}, {aa,aa}, {aa,ab}, {ab,ab}.
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 1;
0, 4, 2, 1;
0, 10, 7, 2, 1;
0, 26, 18, 7, 2, 1;
0, 76, 56, 22, 7, 2, 1;
0, 232, 168, 68, 22, 7, 2, 1;
0, 764, 543, 218, 73, 22, 7, 2, 1;
0, 2620, 1792, 721, 234, 73, 22, 7, 2, 1;
0, 9496, 6187, 2438, 791, 240, 73, 22, 7, 2, 1;
...
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MAPLE
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g:= proc(n) option remember; `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, x^n,
add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..15);
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MATHEMATICA
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g[n_] := g[n] = If[n < 2, 1, g[n - 1] + (n - 1)*g[n - 2]] ;
b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, x^n, Sum[Binomial[g[i] + j - 1, j]*b[n - i*j, i - 1]*x^j, {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jun 04 2018, from Maple *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000085 (for n>0), A294004, A294005, A294006, A294007, A294008, A294009, A294010, A294011, A294012.
Row sums give: A293110.
T(2n,n) gives A293809.
Cf. A293815.
Sequence in context: A140649 A290222 A327549 * A327805 A276689 A091453
Adjacent sequences: A293805 A293806 A293807 * A293809 A293810 A293811
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, Oct 16 2017
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STATUS
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approved
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