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

0,6

Comments

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

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

Row sums give: A000041.

Links

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

Formula

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

Example

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;
  ...

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

Crossrefs

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

Sequence in context: A035213 A340224 A083901 * A355583 A368543 A359302

Adjacent sequences: A274514 A274515 A274516 * A274518 A274519 A274520

Keyword

nonn,tabf

Author

Geoffrey Critzer, Jun 25 2016

Status

approved