A382871 -id:A382871 - cp's OEIS frontend

A382955 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n * y^k] Product_{p prime} (1 + x^p + y^p).

1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 2, 0, 2, 1, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 1, 0, 2, 0, 2, 0, 1, 1, 0, 2

Offset: 0

Seiichi Manyama, Apr 10 2025

			Square array begins:
  1, 0, 1, 1, 0, 2, 0, 2, 1, 1, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  1, 0, 0, 1, 0, 1, 0, 1, 1, 0, ...
  1, 0, 1, 0, 0, 1, 0, 2, 0, 1, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  2, 0, 1, 1, 0, 2, 0, 2, 0, 1, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  2, 0, 1, 2, 0, 2, 0, 2, 1, 0, ...
  1, 0, 1, 0, 0, 0, 0, 1, 0, 1, ...
  1, 0, 0, 1, 0, 1, 0, 0, 1, 0, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..1 give A000586, A000004.
Main diagonal gives 2*A382871(n) (for n > 0).
Cf. A284593, A382956.

A(n,k) = A(k,n).

A382954 Number of ways to partition distinct prime numbers into three disjoint sets such that the sum of each set equals n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 3, 2, 8, 1, 1, 4, 0, 14, 9, 1, 4, 7, 16, 26, 31, 17, 3, 19, 39, 54, 20, 62, 9, 41, 96, 89, 62, 66, 34, 59, 197, 241, 289, 69, 124, 184, 133, 481, 440, 148, 225, 394, 709, 808, 984, 555, 414, 799

Offset: 0

Seiichi Manyama, Apr 10 2025

nonn

Conjecture: a(n) > 0 for n > 35.

			a(29) = 3: [29; 19, 7, 3; 13, 11, 5], [29; 17, 7, 3, 2; 13, 11, 5], [29; 17, 7, 5; 13, 11, 3, 2].
a(30) = 2: [23, 7; 19, 11; 17, 13], [23, 5, 2; 19, 11; 17, 13].

Cf. A000607, A258281, A382871.

a(n) = 1/6 * [(x*y*z)^n] Product_{p prime} (1 + x^p + y^p + z^p) for n > 0.

cp's OEIS Frontend

A382955 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n * y^k] Product_{p prime} (1 + x^p + y^p).

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