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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). 0
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified August 15 02:47 EDT 2022. Contains 356122 sequences. (Running on oeis4.)