A282516 Number T(n,k) of k-element subsets of [n] having a prime element sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 2, 4, 1, 0, 0, 3, 5, 2, 2, 0, 0, 3, 7, 6, 4, 2, 0, 0, 4, 9, 10, 11, 7, 1, 0, 0, 4, 11, 18, 21, 13, 7, 2, 0, 0, 4, 14, 26, 34, 31, 20, 7, 3, 0, 0, 4, 18, 37, 53, 59, 51, 32, 11, 2, 0, 0, 5, 21, 47, 82, 110, 117, 85, 35, 12, 2, 0

Offset: 0

Author

Alois P. Heinz, Feb 17 2017

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  0;
  0, 0;
  0, 1,  1;
  0, 2,  2,  0;
  0, 2,  4,  1,  0;
  0, 3,  5,  2,  2,   0;
  0, 3,  7,  6,  4,   2,   0;
  0, 4,  9, 10, 11,   7,   1,  0;
  0, 4, 11, 18, 21,  13,   7,  2,  0;
  0, 4, 14, 26, 34,  31,  20,  7,  3,  0;
  0, 4, 18, 37, 53,  59,  51, 32, 11,  2, 0;
  0, 5, 21, 47, 82, 110, 117, 85, 35, 12, 2, 0;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Columns k=0-10 give: A000004, A000720, A071917, A320678, A320679, A320680, A320681, A320682, A320683, A320684, A320685.

Row sums give A127542.

Main diagonal gives A185012.

First lower diagonal gives A282518.

T(2n,n) gives A282517.

Programs

Maple

b:= proc(n, s) option remember; expand(`if`(n=0,
      `if`(isprime(s), 1, 0), b(n-1, s)+x*b(n-1, s+n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..16);

Mathematica

b[n_, s_] := b[n, s] = Expand[If[n==0, If[PrimeQ[s], 1, 0], b[n-1, s] + x*b[n-1, s+n]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Mar 21 2017, translated from Maple *)