A395313 Triangle read by rows: T(n, k) is the number of partitions of n into distinct parts with exactly k prime parts, where 0 <= k <= A350174(n).

1, 1, 0, 1, 0, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 2, 2, 3, 3, 3, 3, 3, 1, 2, 6, 3, 1, 2, 8, 4, 1, 3, 8, 6, 1, 4, 8, 8, 2, 5, 10, 9, 3, 5, 12, 11, 4, 4, 17, 13, 3, 1, 6, 18, 16, 5, 1, 8, 18, 20, 8, 0, 8, 22, 24, 10, 0, 9, 28, 26, 11, 2, 11, 30, 32, 14, 2

Offset

0,6

Comments

This triangle refines A000009 by the number of prime parts.

Links

Felix Huber, Table of n, a(n) for n = 0..10016

Formula

G.f.: F(x, y) = Product_{m>=1} (1 + x^m)*Product_{p in A000040} (1 + (y - 1)*x^p).

T(n, k) = [x^n*y^k] Product_{m>=1} (1 + x^m)*Product_{p in A000040} (1 + (y - 1)*x^p).

Sum_{k>=0} T(n, k) = A000009(n).

Example

The triangle begins:
  n\k |  0   1   2   3   4
  ----+-------------------
   0  |  1
   1  |  1
   2  |  0   1
   3  |  0   2
   4  |  1   1
   5  |  1   1   1
   6  |  1   2   1
   7  |  1   3   1
   8  |  1   3   2
   9  |  2   3   3
  10  |  3   3   3   1
  11  |  2   6   3   1
  12  |  2   8   4   1
  13  |  3   8   6   1
  14  |  4   8   8   2
  15  |  5  10   9   3
  16  |  5  12  11   4
  17  |  4  17  13   3   1
  18  |  6  18  16   5   1
  19  |  8  18  20   8   0
  20  |  8  22  24  10   0
  21  |  9  28  26  11   2
  22  | 11  30  32  14   2
T(6, 0) = 1: [6]; T(6, 1) = 2: [1, 5], [2, 4]; T(6, 2) = 1: [1, 2, 3].

Maple

A350174 := proc(n) option remember;
    local m, s;
    m := 0; s := 0;
    while s + ithprime(m + 1) <= n do
        m := m + 1;
        s := s + ithprime(m);
    end do;
    m;
end proc:
T := proc(n, k, i := n) option remember;
    `if`(n = 0 and k = 0, 1,
    `if`(i <= 0, 0,
    `if`(isprime(i), T(n, k, i - 1) + T(n - i, k - 1, i - 1), T(n, k, i - 1) + T(n - i, k, i - 1))));
end proc:
seq(seq(T(n, k), k = 0 .. A350174(n)), n = 0 .. 22);

Crossrefs

Cf. A000009, A000040, A350174, A395211.

Sequence in context: A367760 A195007 A276947 * A178085 A078572 A122750

Adjacent sequences: A395310 A395311 A395312 * A395314 A395315 A395316

Keyword

nonn,tabf

Author

Felix Huber, May 01 2026

Status

approved