A336676 Irregular triangle read by rows: T(n, k) is the number of self-conjugate partitions of n into exactly k parts, for n >= 0 and 2*k - 1 <= n <= k^2.

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 0, 2, 2, 2, 1, 1, 0, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 3, 2, 2, 1, 1, 0, 1, 2, 2, 2, 2, 1

Offset

0,40

Comments

If n > 0, T(n, k) is the number of self-conjugate partitions of n-2*k+1 into fewer than k parts. Also, the number of partitions of n into distinct odd parts with largest part 2*k-1.

Columns are symmetric, for k > 0: T(n, k) = T(k^2 + 2*k - 1 - n, k).

Within the range 2*k - 1 <= n <= k^2, T(n, k) = 0 iff n = 2*k + 1 or n = k^2 - 2.

Aligning columns k > 0 to the top (shifting each by 1-2k positions) and transposing gives A178666.

Links

Álvar Ibeas, Rows until n=399, flattened

Álvar Ibeas, Rows until n=17

Formula

T(0, 0) = 1. If n > 0, T(n, k) = Sum_{i < k} T(n - 2*k + 1, i) = A178666(k - 2, n - 2*k + 1).

Column g.f., for k > 0: x^(2*k - 1) * Product_{i=1,...,k-1} (1 + x^(2*i-1)).

If n < 4*k - 2, T(n, k) = A000700(n - 2*k + 1).

Crossrefs

Cf. A178666, A000700 (row sums), A011782 (column sums).

Sequence in context: A236747 A194285 A367623 * A135341 A344299 A033665

Adjacent sequences: A336673 A336674 A336675 * A336677 A336678 A336679

Keyword

nonn,tabf

Author

Álvar Ibeas, Jul 30 2020

Status

approved