A204459 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of k-element subsets that can be chosen from {1,2,...,k*n} having element sum k*(k*n+1)/2.

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 2, 1, 1, 0, 1, 0, 3, 0, 1, 0, 1, 8, 8, 4, 1, 1, 0, 1, 0, 33, 0, 5, 0, 1, 0, 1, 58, 141, 86, 25, 6, 1, 1, 0, 1, 0, 676, 0, 177, 0, 7, 0, 1, 0, 1, 526, 3370, 3486, 1394, 318, 50, 8, 1, 1, 0, 1, 0, 17575, 0, 11963, 0, 519, 0, 9, 0, 1

Offset: 0

Author

Alois P. Heinz, Jan 15 2012

Keywords

• nonn
• tabl

Comments

A(n,k) is the number of partitions of k*(k*n+1)/2 into k distinct parts <=k*n.

A(n,k) = 0 if k>0 and (n = 0 or k*(k*n+1) mod 2 = 1).

Examples

			A(0,0) = 1: {}.
A(1,1) = 1: {1}.
A(5,1) = 1: {3}.
A(1,5) = 1: {1,2,3,4,5}.
A(2,2) = 2: {1,4}, {2,3}.
A(3,2) = 3: {1,6}, {2,5}, {3,4}.
A(2,3) = 0: no subset of {1,2,3,4,5,6} has element sum 3*(3*2+1)/2 = 21/2.
A(4,2) = 4: {1,8}, {2,7}, {3,6}, {4,5}.
A(3,3) = 8: {1,5,9}, {1,6,8}, {2,4,9}, {2,5,8}, {2,6,7}, {3,4,8}, {3,5,7}, {4,5,6}.
A(2,4) = 8: {1,2,7,8}, {1,3,6,8}, {1,4,5,8}, {1,4,6,7}, {2,3,5,8}, {2,3,6,7}, {2,4,5,7}, {3,4,5,6}.
Square array A(n,k) begins:
  1, 0, 0,  0,   0,    0,     0,      0, ...
  1, 1, 1,  1,   1,    1,     1,      1, ...
  1, 0, 2,  0,   8,    0,    58,      0, ...
  1, 1, 3,  8,  33,  141,   676,   3370, ...
  1, 0, 4,  0,  86,    0,  3486,      0, ...
  1, 1, 5, 25, 177, 1394, 11963, 108108, ...
  1, 0, 6,  0, 318,    0, 32134,      0, ...
  1, 1, 7, 50, 519, 5910, 73294, 957332, ...
		

Links

• Alois P. Heinz, Antidiagonals d=0..60

Crossrefs

Rows n=0-10 give: A000007, A000012, A063074, A109655, A204460, A204461, A204462, A204463, A204464, A204465, A204466.

Columns k=0..10 give: A000012, A000035, A001477, A204467, A204468, A204469, A204470, A204471, A204472, A204473, A204474.

Main diagonal gives: A052456.

Programs

Maple

b:= proc(n, i, t) option remember;
      `if`(it*(2*i-t+1)/2, 0,
      `if`(n=0, 1, b(n, i-1, t) +`if`(n
				

Mathematica

b[n_, i_, t_] /; it*((2*i-t+1)/2) = 0; b[0, , ] = 1; b[n_, i_, t_] := b[n, i, t] = b[n, i-1, t] + If[n, 0] = 1; a[0, ] = 0; a[n_, k_] := With[{s = k*(k*n+1)}, If[Mod[s, 2] == 1, 0, b[s/2, k*n, k]]]; Flatten[ Table[ a[n, d-n], {d, 0, 15}, {n, 0, d}]] (* Jean-François Alcover, Jun 15 2012, translated from Maple, after Alois P. Heinz *)