A180281 Triangle read by rows: T(n,k) = number of arrangements of n indistinguishable balls in n boxes with the maximum number of balls in any box equal to k.

1, 1, 2, 1, 6, 3, 1, 18, 12, 4, 1, 50, 50, 20, 5, 1, 140, 195, 90, 30, 6, 1, 392, 735, 392, 147, 42, 7, 1, 1106, 2716, 1652, 672, 224, 56, 8, 1, 3138, 9912, 6804, 2970, 1080, 324, 72, 9, 1, 8952, 35850, 27600, 12825, 4950, 1650, 450, 90, 10, 1, 25652, 128865, 110715, 54450, 22022, 7865, 2420, 605, 110, 11

Offset

1,3

Comments

To clarify a slight ambiguity in the definition, the heaviest box in such an arrangement should contain exactly k balls. - Gus Wiseman, Sep 22 2016

Links

Alois P. Heinz, Rows n = 1..200, flattened (first 59 rows from R. H. Hardin)

Formula

Empirical: right half of table, T(n,k) = n*binomial(2*n-k-2,n-2) for 2*k > n; also, T(n,2) = Sum_{j=1..n} binomial(n,j)*binomial(n-j,j) = 2*A097861(n). - Robert Gerbicz in the Sequence Fans Mailing List

From Alois P. Heinz, Aug 17 2018: (Start)

T(n,k) = [x^n] ((x^(k+1)-1)/(x-1))^n - ((x^k-1)/(x-1))^n.

T(n,k) = A305161(n,k) - A305161(n,k-1). (End)

Example

The T(4,2)=18 arrangements are {0022, 0112, 0121, 0202, 0211, 0220, 1012, 1021, 1102, 1120, 1201, 1210, 2002, 2011, 2020, 2101, 2110, 2200}.
Triangle starts
  1
  1   2
  1   6   3
  1  18  12  4
  1  50  50 20  5
  1 140 195 90 30 6
  ...

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0, 0, add(b(n-j, i-1, k), j=0..min(n, k))))
    end:
T:= (n, k)-> b(n$2, k)-b(n$2, k-1):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Aug 16 2018
# second Maple program:
T:= (n, k)-> coeff(series(((x^(k+1)-1)/(x-1))^n
             -((x^k-1)/(x-1))^n, x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Aug 17 2018

Mathematica

T[n_, k_]:=Select[Tuples[Range[0, k], n], And[Max[#]===k, Total[#]===n]&]; (* Gus Wiseman, Sep 22 2016 *)
SequenceForm@@@T[4, 2] (* example *)
Join@@Table[Length[T[n, k]], {n, 1, 6}, {k, 1, n}] (* sequence *)
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, 0, Sum[b[n-j, i-1, k], {j, 0, Min[n, k]}]]];
T[n_, k_] := b[n, n, k] - b[n, n, k-1];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Aug 28 2022, after Alois P. Heinz *)

Crossrefs

Cf. A097861, A180282, A180291.

Row sums give A088218.

T(n,ceiling(n/2)) gives A318160.

T(2n,n) gives A318161.

T(2n-1,n) gives A318161.

Sequence in context: A121468 A168151 A213221 * A187888 A239102 A239103

Adjacent sequences: A180278 A180279 A180280 * A180282 A180283 A180284

Keyword

nonn,tabl

Author

R. H. Hardin, Aug 24 2010

Status

approved