A396143 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of strict integer partitions of 2*n with reverse-alternating sum -2*k.

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 2, 1, 1, 1, 0, 0, 2, 2, 1, 1, 1, 0, 0, 3, 3, 2, 1, 1, 1, 0, 0, 3, 5, 3, 2, 1, 1, 1, 0, 0, 4, 6, 5, 3, 2, 1, 1, 1, 0, 0, 4, 8, 7, 5, 3, 2, 1, 1, 1, 0, 0, 5, 10, 10, 7, 5, 3, 2, 1, 1, 1, 0, 0, 5, 13, 13, 10, 7, 5, 3, 2, 1, 1, 1, 0

Offset

0,17

Comments

Also the number of partitions of n+2*k^2 into 2*k distinct parts not containing the part 2*k, except when n = k = 0.

Links

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

Formula

G.f. of column k: Sum_{i=1..2*k} q^(i^2+k) * q_binomial(2*k-1,i-1) / Product_{j=1..i} (1-q^j) for k > 0.

T(n,k) + A344649(n,k) = A152146(n,k), except when n = k = 0.

Example

Triangle begins:
  1;
  0, 0;
  0, 1, 0;
  0, 1, 1, 0;
  0, 1, 1, 1, 0;
  0, 2, 1, 1, 1, 0;
  0, 2, 2, 1, 1, 1, 0;
  0, 3, 3, 2, 1, 1, 1, 0;
  0, 3, 5, 3, 2, 1, 1, 1, 0;
  0, 4, 6, 5, 3, 2, 1, 1, 1, 0;
  0, 4, 8, 7, 5, 3, 2, 1, 1, 1, 0;

Crossrefs

Columns k=3..4 give A396144, A396145.

Cf. A008284, A079126, A152146, A344649.

Sequence in context: A188921 A334568 A072617 * A348541 A210825 A056226

Adjacent sequences: A396140 A396141 A396142 * A396144 A396145 A396146

Keyword

nonn,tabl

Author

Seiichi Manyama, May 18 2026

Status

approved