A341415 Triangle read by rows: T(n,k) is the number of grand Dyck paths of semilength n having degree of symmetry k (n >= 0, 0 <= k <= n).

1, 0, 2, 2, 0, 4, 4, 8, 0, 8, 14, 16, 24, 0, 16, 44, 64, 48, 64, 0, 32, 148, 208, 216, 128, 160, 0, 64, 504, 736, 720, 640, 320, 384, 0, 128, 1750, 2592, 2672, 2176, 1760, 768, 896, 0, 256, 6156, 9280, 9696, 8448, 6080, 4608, 1792, 2048, 0, 512

Offset

0,3

Comments

The degree of symmetry of a grand Dyck path is defined as the number of steps in the first half that are mirror images of steps in the second half, with respect to the reflection along a vertical line through the midpoint of the path.

Links

Table of n, a(n) for n=0..54.

Sergi Elizalde, The degree of symmetry of lattice paths, arXiv:2002.12874 [math.CO], 2020.

Sergi Elizalde, Measuring symmetry in lattice paths and partitions, Sem. Lothar. Combin. 84B.26, 12 pp (2020).

Formula

G.f.: 1/(2(1-u)z+sqrt(1-4z)).

Example

For n=3 there are 4 grand Dyck paths with degree of symmetry equal to 0, namely uddduu, uudddu, duuudd, dduuud.
The triangle begins:
    1
    0    2
    2    0    4
    4    8    0    8
   14   16   24    0   16
   44   64   48   64    0   32
  148  208  216  128  160    0  64
  504  736  720  640  320  384   0  128

Crossrefs

Cf. A000079 (diagonal), A000984 (row sums).

Sequence in context: A342472 A308720 A086882 * A328141 A168587 A320119

Adjacent sequences: A341412 A341413 A341414 * A341416 A341417 A341418

Keyword

nonn,tabl

Author

Sergi Elizalde, Feb 12 2021

Status

approved