A293808 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.

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

Offset

0,5

Links

Alois P. Heinz, Rows n = 0..200, flattened

Index entries for triangles generated by the Multiset Transformation

Formula

G.f.: Product_{j>=1} 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(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;
  ...

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 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);

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 || 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 *)

Crossrefs

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

Keyword

nonn,tabl

Author

Alois P. Heinz, Oct 16 2017

Status

approved