A215520 Number T(n,k) of distinct values of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n with largest part <= k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 5, 6, 7, 1, 4, 7, 9, 10, 11, 1, 4, 8, 10, 12, 13, 14, 1, 5, 10, 14, 17, 18, 19, 20, 1, 5, 12, 16, 21, 23, 25, 26, 27, 1, 6, 14, 20, 27, 29, 32, 34, 35, 36, 1, 6, 16, 22, 32, 35, 40, 43, 45, 46, 47, 1, 7, 19, 28, 40, 45, 52, 57, 60, 62, 63, 64

Offset

1,3

Comments

Differs from A026820 first at T(7,4).

Differs from A213008 first at T(8,3).

Links

Alois P. Heinz, Rows n = 1..75, flattened

Wikipedia, Multinomial coefficients

Example

T(3,2) =  2 = |{3!/(2!*1!), 3!/(1!*1!*1!)}| = |{3, 6}|.
T(5,2) =  3 = |{30, 60, 120}|.
T(7,4) = 10 = |{35, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040}|.
T(8,3) = 10 = |{560, 1120, 1680, 2520, 3360, 5040, 6720, 10080, 20160, 40320}|.
T(9,2) =  5 = |{22680, 45360, 90720, 181440, 362880}|.
Triangle T(n,k) begins:
  1;
  1,  2;
  1,  2,  3;
  1,  3,  4,  5;
  1,  3,  5,  6,  7;
  1,  4,  7,  9, 10, 11;
  1,  4,  8, 10, 12, 13, 14;
  1,  5, 10, 14, 17, 18, 19, 20;
  1,  5, 12, 16, 21, 23, 25, 26, 27;
  1,  6, 14, 20, 27, 29, 32, 34, 35, 36;

Maple

b:= proc(n, i) option remember; `if`(n=0, {1}, `if`(i<1, {},
      {b(n, i-1)[], seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=1..n/i)}))
    end:
T:= (n, k)-> nops(b(n, k)):
seq(seq(T(n, k), k=1..n), n=1..14);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, Join[b[n, i - 1], Table[ b[n - i*j, i - 1] *i!^j, {j, 1, n/i}] // Flatten]] // Union]; T[n_, k_] := Length[b[n, k]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 21 2015, after Alois P. Heinz *)

Crossrefs

Main diagonal gives: A070289.

Columns k=1-3 give: A000012, A008619 (for n>1), A001399 (for n>2).

Cf. A026820, A213008, A215521.

Sequence in context: A209556 A109974 A213008 * A026820 A330661 A091438

Adjacent sequences: A215517 A215518 A215519 * A215521 A215522 A215523

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 14 2012

Status

approved