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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).

%I A341415 #31 Feb 12 2021 18:43:43
%S A341415 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,
%T A341415 160,0,64,504,736,720,640,320,384,0,128,1750,2592,2672,2176,1760,768,
%U A341415 896,0,256,6156,9280,9696,8448,6080,4608,1792,2048,0,512
%N 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).
%C A341415 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.
%H A341415 Sergi Elizalde, <a href="https://arxiv.org/abs/2002.12874">The degree of symmetry of lattice paths</a>, arXiv:2002.12874 [math.CO], 2020.
%H A341415 Sergi Elizalde, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2020/26.html">Measuring symmetry in lattice paths and partitions</a>, Sem. Lothar. Combin. 84B.26, 12 pp (2020).
%F A341415 G.f.: 1/(2(1-u)z+sqrt(1-4z)).
%e A341415 For n=3 there are 4 grand Dyck paths with degree of symmetry equal to 0, namely uddduu, uudddu, duuudd, dduuud.
%e A341415 The triangle begins:
%e A341415     1
%e A341415     0    2
%e A341415     2    0    4
%e A341415     4    8    0    8
%e A341415    14   16   24    0   16
%e A341415    44   64   48   64    0   32
%e A341415   148  208  216  128  160    0  64
%e A341415   504  736  720  640  320  384   0  128
%Y A341415 Cf. A000079 (diagonal), A000984 (row sums).
%K A341415 nonn,tabl
%O A341415 0,3
%A A341415 _Sergi Elizalde_, Feb 12 2021