A305161 Number A(n,k) of compositions of n into exactly n nonnegative parts <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 3, 7, 1, 0, 1, 1, 3, 10, 19, 1, 0, 1, 1, 3, 10, 31, 51, 1, 0, 1, 1, 3, 10, 35, 101, 141, 1, 0, 1, 1, 3, 10, 35, 121, 336, 393, 1, 0, 1, 1, 3, 10, 35, 126, 426, 1128, 1107, 1, 0, 1, 1, 3, 10, 35, 126, 456, 1520, 3823, 3139, 1, 0

Offset

0,13

Links

Alois P. Heinz, Antidiagonals n = 0..200

Formula

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

A(n,k) - A(n,k-1) = A180281(n,k) for n,k > 0.

A(n,k) = A(n,n) for all k >= n.

Example

A(3,1) = 1: 111.
A(3,2) = 7: 012, 021, 102, 111, 120, 201, 210.
A(3,3) = 10: 003, 012, 021, 030, 102, 111, 120, 201, 210, 300.
A(4,2) = 19: 0022, 0112, 0121, 0202, 0211, 0220, 1012, 1021, 1102, 1111, 1120, 1201, 1210, 2002, 2011, 2020, 2101, 2110, 2200.
A(4,3) = 31: 0013, 0022, 0031, 0103, 0112, 0121, 0130, 0202, 0211, 0220, 0301, 0310, 1003, 1012, 1021, 1030, 1102, 1111, 1120, 1201, 1210, 1300, 2002, 2011, 2020, 2101, 2110, 2200, 3001, 3010, 3100.
Square array A(n,k) begins:
  1, 1,    1,    1,    1,    1,    1,    1,    1, ...
  0, 1,    1,    1,    1,    1,    1,    1,    1, ...
  0, 1,    3,    3,    3,    3,    3,    3,    3, ...
  0, 1,    7,   10,   10,   10,   10,   10,   10, ...
  0, 1,   19,   31,   35,   35,   35,   35,   35, ...
  0, 1,   51,  101,  121,  126,  126,  126,  126, ...
  0, 1,  141,  336,  426,  456,  462,  462,  462, ...
  0, 1,  393, 1128, 1520, 1667, 1709, 1716, 1716, ...
  0, 1, 1107, 3823, 5475, 6147, 6371, 6427, 6435, ...

Maple

A:= (n, k)-> coeff(series(((x^(k+1)-1)/(x-1))^n, x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
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:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

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]}]]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 05 2019, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A000007, A000012, A002426, A005725, A187925, A318113, A318114, A318115, A318116, A167403, A318117.

Rows n=0-1 give: A000012, A057427.

Main diagonal gives A088218 or A001700(n-1) for n>0.

A(n+1,n) gives A048775.

Cf. A180281.

Sequence in context: A226873 A293960 A062719 * A250484 A294250 A117417

Adjacent sequences: A305158 A305159 A305160 * A305162 A305163 A305164

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 17 2018

Status

approved