A355011 Array read by ascending antidiagonals: T(n, k) is the number of self-conjugate n-core partitions with k corners.

1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 3, 4, 5, 1, 1, 3, 9, 5, 7, 1, 1, 4, 9, 15, 7, 8, 1, 1, 4, 16, 15, 27, 8, 10, 1, 1, 5, 16, 34, 27, 37, 10, 11, 1, 1, 5, 25, 34, 76, 37, 55, 11, 13, 1, 1, 6, 25, 65, 76, 124, 55, 69, 13, 14, 1, 1, 6, 36, 65, 175, 124, 216, 69, 93, 14, 16, 1, 1

Offset

2,4

Comments

T(n, k) is also equal to the number of cornerless symmetric Motzkin paths of length 2*k + n - 1 with n - 1 flat steps (see Theorem 3.7 and Proposition 3.8 at pp. 16 - 17 in Cho et al.).

Links

Table of n, a(n) for n=2..79.

Hyunsoo Cho, JiSun Huh, Hayan Nam, and Jaebum Sohn, Combinatorics on bounded free Motzkin paths and its applications, arXiv:2205.15554 [math.CO], 2022.

Formula

T(n, k) = Sum_{i=1..min(k,floor(n/2))} binomial(floor((k-1)/2), floor((i-1)/2))*binomial(floor(k/2), floor(i/2))*binomial(floor(n/2)+k-i, k). (See proposition 3.8 in Cho et al.).

T(4, n) = T(5, n) = A001651(n+1).

Example

The array begins:
  1,  1,  1,   1,   1,   1,    1,    1, ...
  1,  1,  1,   1,   1,   1,    1,    1, ...
  2,  4,  5,   7,   8,  10,   11,   13, ...
  2,  4,  5,   7,   8,  10,   11,   13, ...
  3,  9, 15,  27,  37,  55,   69,   93, ...
  3,  9, 15,  27,  37,  55,   69,   93, ...
  4, 16, 34,  76, 124, 216,  309,  471, ...
  4, 16, 34,  76, 124, 216,  309,  471, ...
  5, 25, 65, 175, 335, 675, 1095, 1875, ...
  5, 25, 65, 175, 335, 675, 1095, 1875, ...
  ...

Mathematica

T[n_, k_]:=Sum[Binomial[Floor[(k-1)/2], Floor[(i-1)/2]]Binomial[Floor[k/2], Floor[i/2]]Binomial[Floor[n/2]+k-i, k], {i, Min[k, Floor[n/2]]}]; Flatten[Table[T[n-k+1, k], {n, 2, 13}, {k, 1, n-1}]]

Crossrefs

Cf. A000012 (n = 2,3), A001651, A004526 (k = 1), A008794 (k = 2), A247643 (n = 6,7), A355010.

Sequence in context: A136788 A334622 A136450 * A131054 A267998 A265005

Adjacent sequences: A355008 A355009 A355010 * A355012 A355013 A355014

Keyword

nonn,tabl

Author

Stefano Spezia, Jun 15 2022

Status

approved