A291057 -id:A291057 - 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).

A294129 Numbers n for which exactly one length minimal language exists having exactly n nonempty words 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

0, 1, 3, 7, 17, 43, 119, 351, 1115, 3735, 13231, 48927, 189079, 757583, 3148063, 13497599, 59704335, 271503647, 1268817471, 6078518911, 29837183007, 149789875903, 768674514815, 4026518397439, 21518708975039, 117199152735615, 650184360936191, 3670861106911743

Offset: 1

Alois P. Heinz, Oct 23 2017

nonn

Numbers n such that A291057(n) equals 1.

Numbers n such that the smallest nonzero term in column n of A293815 equals 1.

			0 is a term because there is only one length minimal language with 0 words: {}.
1 is a term: {a}.
3 is a term: {a, aa, ab}.
7 is a term: {a, aa, ab, aaa, aab, aba, abc}.
17 is a term: {a, aa, ab, aaa, aab, aba, abc, aaaa, aaab, aaba, aabb, aabc, abaa, abab, abac, abca, abcd}.

Alois P. Heinz, Table of n, a(n) for n = 1..800

Cf. A000085, A245176, A291057, A293815.

a:= proc(n) option remember; `if`(n<4, n*(n-1)/2,
      2*a(n-1)+(n-3)*a(n-2)-(n-2)*a(n-3))
    end:
seq(a(n), n=1..30);

a(n) = a(n-1) + A000085(n-1) for n>1, a(1) = 0.
a(n) = 2*a(n-1)+(n-3)*a(n-2)-(n-2)*a(n-3) for n>= 4, a(n) = n*(n-1)/2 for n<4.
a(n) = A245176(n-1) - 1 for n>0.

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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