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A293815
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.
14
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
OFFSET
0,5
COMMENTS
The smallest nonzero term in column k is A291057(k).
LINKS
FORMULA
G.f.: Product_{j>=1} (1+y*x^j)^A000085(j).
EXAMPLE
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;
...
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; 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);
MATHEMATICA
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]];
Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jun 04 2018, from Maple *)
CROSSREFS
Columns k=0-10 give: A000007, A000085 (for n>0), A293964, A293965, A293966, A293967, A293968, A293969, A293970, A293971, A293972.
Row sums give A293114.
Sequence in context: A081880 A376923 A144289 * A373691 A339941 A211318
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Oct 16 2017
STATUS
approved