A294129 -id:A294129 - cp's OEIS frontend

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.

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

Alois P. Heinz, Oct 16 2017

The smallest nonzero term in column k is A291057(k).

			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;
  ...

Alois P. Heinz, Rows n = 0..300, flattened

Columns k=0-10 give: A000007, A000085 (for n>0), A293964, A293965, A293966, A293967, A293968, A293969, A293970, A293971, A293972.
Row sums give A293114.
Cf. A208741, A293808, A291057, A294129.

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);

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 *)

G.f.: Product_{j>=1} (1+y*x^j)^A000085(j).

A291057 Cardinality of the smallest nonempty class of length minimal languages with exactly n nonempty words each over a countably infinite alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 4, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 26, 325, 2600, 14950, 65780, 230230, 657800, 1562275, 3124550, 5311735, 7726160, 9657700, 10400600, 9657700, 7726160, 5311735, 3124550, 1562275, 657800, 230230, 65780, 14950, 2600, 325, 26, 1

Offset: 0

Alois P. Heinz, Oct 20 2017

a(n) is the smallest nonzero term in column n of A293815.

			a(0) = 1: {{}}.
a(1) = 1: {{a}}.
a(2) = 2: {{a,aa}, {a,ab}}.
a(3) = 1: {{a,aa,ab}}.
a(4) = 4: {{a,aa,ab,aaa}, {a,aa,ab,aab}, {a,aa,ab,aba}, {a,aa,ab,abc}}.
a(5) = 6: {{a,aa,ab,aaa,aab}, {a,aa,ab,aaa,aba}, {a,aa,ab,aaa,abc}, {a,aa,aab,aba}, {a,aa,ab,aab,ab,abc}, {a,aa,ab,aba,abc}}.
a(6) = 4: {{a,aa,ab,aaa,aab,aba}, {a,aa,ab,aaa,aab,abc}, {a,aa,ab,aaa,aba,abc}, {a,aa,ab,aab,aba,abc}}.
a(7) = 1: {{a,aa,ab,aaa,aab,aba,abc}}.
Breaking the sequence into lines after each 1 gives an irregular triangle whose j-th row equals the A000085(j)-th row of A007318 without its leftmost term. The leftmost column of this triangle is A000085:
   1;
   1;
   2,   1;
   4,   6,    4,     1;
  10,  45,  120,   210,   252,    210,    120,      45,      10,   1;
  26, 325, 2600, 14950, 65780, 230230, 657800, 1562275, 3124550, ...
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..1115

Cf. A000085, A007318, A293815, A294129.

cp's OEIS Frontend

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.

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