A274517 Number T(n,k) of integer partitions of n with exactly k distinct primes.

1, 1, 1, 1, 1, 2, 2, 3, 2, 4, 1, 3, 7, 1, 3, 9, 3, 5, 12, 5, 6, 15, 9, 8, 22, 11, 1, 8, 28, 19, 1, 12, 38, 24, 3, 13, 46, 38, 4, 17, 62, 48, 8, 19, 77, 68, 12, 26, 98, 87, 20, 28, 117, 127, 24, 1, 37, 152, 154, 41, 1, 40, 183, 210, 55, 2, 52, 230, 260, 82, 3

Offset: 0

Author

Geoffrey Critzer, Jun 25 2016

Keywords

• nonn
• tabf

Comments

Row lengths increase by 1 at row A007504(n).

Columns k=0-1 give: A002095, A132381.

Row sums give: A000041.

Examples

			T(6,1) = 7 because we have: 5+1, 4+2, 3+3, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1+1.
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  2;
  2,  3;
  2,  4,  1;
  3,  7,  1;
  3,  9,  3;
  5, 12,  5;
  6, 15,  9;
  8, 22, 11, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..1000, flattened

Crossrefs

Cf. A000041, A002095, A007504, A132381, A222656.

Programs

Maple

b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
      `if`(j>0 and isprime(i), x, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Jun 26 2016

Mathematica

nn = 20; Map[Select[#, # > 0 &] &, CoefficientList[Series[Product[
      1/(1 - z^k), {k,Select[Range[1000], PrimeQ[#] == False &]}] Product[
      u/(1 - z^j) - u + 1, {j, Table[Prime[n], {n, 1, nn}]}], {z, 0,
     nn}], {z, u}]] // Grid

Formula

G.f.: Product_{k>=1} (1 - x^prime(k))/(1 - x^k)*(y/(1-x^prime(k)) - y + 1).