A361521 Array read by descending antidiagonals. A(n, k) is the number of the nonempty multiset combinations of {0, 1} as defined in A361682.

0, 0, 0, 0, 2, 0, 0, 5, 4, 0, 0, 9, 12, 6, 0, 0, 14, 24, 21, 8, 0, 0, 20, 40, 45, 32, 10, 0, 0, 27, 60, 78, 72, 45, 12, 0, 0, 35, 84, 120, 128, 105, 60, 14, 0, 0, 44, 112, 171, 200, 190, 144, 77, 16, 0, 0, 54, 144, 231, 288, 300, 264, 189, 96, 18, 0

Offset: 0

Author

Peter Luschny, Mar 22 2023

Keywords

• nonn
• tabl

Comments

A detailed combinatorial interpretation can be found in A361682.

Examples

			[0] 0,  0,  0,   0,   0,   0,   0,    0, ...  A000004
[1] 0,  2,  5,   9,  14,  20,  27,   35, ...  A000096
[2] 0,  4, 12,  24,  40,  60,  84,  112, ...  A046092
[3] 0,  6, 21,  45,  78, 120, 171,  231, ...  A081266
[4] 0,  8, 32,  72, 128, 200, 288,  392, ...  A139098
[5] 0, 10, 45, 105, 190, 300, 435,  595, ...
[6] 0, 12, 60, 144, 264, 420, 612,  840, ...  A153792
[7] 0, 14, 77, 189, 350, 560, 819, 1127, ...
       | A028347 |     A163761
     A005843  A067725
.
[0] 0;
[1] 0,  0;
[2] 0,  2,   0;
[3] 0,  5,   4,   0;
[4] 0,  9,  12,   6,   0;
[5] 0, 14,  24,  21,   8,   0;
[6] 0, 20,  40,  45,  32,  10,   0;
[7] 0, 27,  60,  78,  72,  45,  12,  0;
[8] 0, 35,  84, 120, 128, 105,  60, 14,  0;
[9] 0, 44, 112, 171, 200, 190, 144, 77, 16, 0;
		

Crossrefs

Rows: A000004, A000096, A046092, A081266, A139098, A153792.

Columns: A000004, A005843, A028347, A067725, A163761, A068380.

Cf. A361682.

Programs

Maple

A := (n, k) -> n*k*(4 + n*(k - 1))/2:
for n from 0 to 7 do seq(A(n, k), k = 0..7) od;

Formula

A(n, k) = n*k*(4 + n*(k - 1))/2.

T(n, k) = k*(n - k)*(4 + k*(n - k - 1))/2.

A(n, k) = A361682(n, k) - 1.