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
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 17 2018
STATUS
approved