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A293815
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Number T(n,k) of sets 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, read by rows.
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14
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1, 0, 1, 0, 2, 0, 4, 2, 0, 10, 5, 0, 26, 18, 1, 0, 76, 52, 8, 0, 232, 168, 30, 0, 764, 533, 114, 4, 0, 2620, 1792, 411, 22, 0, 9496, 6161, 1462, 116, 0, 35696, 22088, 5237, 482, 6, 0, 140152, 81690, 18998, 1966, 48, 0, 568504, 313224, 70220, 7682, 274
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OFFSET
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0,5
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COMMENTS
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The smallest nonzero term in column k is A291057(k).
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LINKS
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FORMULA
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G.f.: Product_{j>=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(4,2) = 5: {a,aaa}, {a,aab}, {a,aba}, {a,abc}, {aa,ab}.
T(5,3) = 1: {a,aa,ab}.
Triangle T(n,k) begins:
1;
0, 1;
0, 2;
0, 4, 2;
0, 10, 5;
0, 26, 18, 1;
0, 76, 52, 8;
0, 232, 168, 30;
0, 764, 533, 114, 4;
0, 2620, 1792, 411, 22;
0, 9496, 6161, 1462, 116;
0, 35696, 22088, 5237, 482, 6;
0, 140152, 81690, 18998, 1966, 48;
...
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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, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*binomial(g(i), j)*x^j, j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(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, 1, If[i<1, 0, Sum[b[n - i*j, i-1]* Binomial[g[i], j]*x^j, {j, 0, n/i}]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, n]];
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CROSSREFS
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Columns k=0-10 give: A000007, A000085 (for n>0), A293964, A293965, A293966, A293967, A293968, A293969, A293970, A293971, A293972.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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