A163181 T(n,k) is the number of weak compositions of k into n parts no greater than (n-1) for n>=1, 0<=k<=n(n-1).

1, 1, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1, 1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1, 1, 6, 21, 56, 126, 252, 456, 756, 1161, 1666, 2247, 2856, 3431, 3906, 4221, 4332, 4221, 3906, 3431

Offset: 1

Author

Geoffrey Critzer, Jul 22 2009

Keywords

• nonn
• tabf

Comments

T(n,k) is the number of length n sequences on an alphabet of {0,1,2,...,n-1} that have a sum of k. Equivalently T(n,k) is the number of functions f:{1,2,...,n}->{0,1,2,...,n-1} such that Sum(f(i)=k, i=1...n).

Row n is also row n of the array of q-nomial coefficients. - Matthew Vandermast, Oct 31 2010

Examples

			T(3,4) = 6 because there are 6 ternary sequences of length three that sum to 4: [0, 2, 2], [1, 1, 2], [1, 2, 1], [2, 0, 2], [2, 1, 1], [2, 2, 0].
		

Links

• Alois P. Heinz, Rows n = 1..32, flattened

Crossrefs

The maximum of row n is in column k=n(n-1)/2 = A000217(n-1).

For q-nomial arrays, see A000012, A007318, A027907, A008287, A035343, A063260, A063265, A171890. See also A181567. - Matthew Vandermast, Oct 31 2010

Programs

Maple

b:= proc(n, k, l) option remember; `if`(k=0, 1,
      `if`(l=0, 0, add(b(n, k-j, l-1), j=0..min(n-1, k))))
    end:
T:= (n, k)-> b(n, k, n):
seq(seq(T(n, k), k=0..n*(n-1)), n=1..8);  # Alois P. Heinz, Feb 21 2013

Mathematica

(*warning very inefficient*) Table[Distribution[Map[Total, Strings[Range[n], n]]], {n, 1, 6}]//Grid
nn=100;Table[CoefficientList[Series[Sum[x^i,{i,0,n-1}]^n,{x,0,nn}],x],{n,1,10}]//Grid (* Geoffrey Critzer, Feb 21 2013*)

Formula

O.g.f. for row n is ((1-x^n)/(1-x))^n. For k<=(n-1), T(n,k) = C(n+k-1,k).