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

0,8

Links

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

Example

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;
  ...

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 *)

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.

Sequence in context: A081417 A133388 A354643 * A158092 A145264 A300333

Adjacent sequences: A282513 A282514 A282515 * A282517 A282518 A282519

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 17 2017

Status

approved