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A339754 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k symmetric vertices (n >= 1, 1 <= k <= n).

%I A339754 #40 Feb 13 2021 19:38:13
%S A339754 1,0,2,0,2,3,0,2,6,6,0,4,12,16,10,0,8,24,40,40,20,0,20,60,104,120,90,
%T A339754 35,0,50,150,270,350,330,210,70,0,140,420,768,1040,1080,840,448,126,0,
%U A339754 392,1176,2184,3080,3468,3108,2128,1008,252
%N A339754 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k symmetric vertices (n >= 1, 1 <= k <= n).
%C A339754 A symmetric vertex is a vertex in the first half of the path (not including the midpoint) that is a mirror image of a vertex in the second half, with respect to reflection along the vertical line through the midpoint of the path.
%H A339754 Alois P. Heinz, <a href="/A339754/b339754.txt">Rows n = 1..200, flattened</a>
%H A339754 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 A339754 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).
%e A339754 For n=5 there are 4 Dyck paths with 2 symmetric vertices: uuuuddddud, uduuuudddd, uuudddudud, ududuuuddd.
%e A339754 Triangle begins:
%e A339754   1;
%e A339754   0,   2;
%e A339754   0,   2,    3;
%e A339754   0,   2,    6,    6;
%e A339754   0,   4,   12,   16,   10;
%e A339754   0,   8,   24,   40,   40,   20;
%e A339754   0,  20,   60,  104,  120,   90,   35;
%e A339754   0,  50,  150,  270,  350,  330,  210,   70;
%e A339754   0, 140,  420,  768, 1040, 1080,  840,  448,  126;
%e A339754   0, 392, 1176, 2184, 3080, 3468, 3108, 2128, 1008, 252;
%e A339754   ...
%p A339754 b:= proc(x, y, v) option remember; expand(
%p A339754       `if`(min(y, v, x-max(y, v))<0, 0, `if`(x=0, 1, (l-> add(add(
%p A339754       `if`(y+i=v+j, z, 1)*b(x-1, y+i, v+j), i=l), j=l))([-1, 1]))))
%p A339754     end:
%p A339754 g:= proc(n) option remember; add(b(n, j$2), j=0..n) end:
%p A339754 T:= (n, k)-> coeff(g(n), z, k):
%p A339754 seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, Feb 12 2021
%t A339754 b[x_, y_, v_] := b[x, y, v] = Expand[If[Min[y, v, x - Max[y, v]] < 0, 0, If[x == 0, 1, Function[l, Sum[Sum[If[y + i == v + j, z, 1]*b[x - 1, y + i, v + j], {i, l}], {j, l}]][{-1, 1}]]]];
%t A339754 g[n_] := g[n] = Sum[b[n, j, j], {j, 0, n}];
%t A339754 T[n_, k_] := Coefficient[g[n], z, k];
%t A339754 Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Feb 13 2021, after _Alois P. Heinz_ *)
%Y A339754 Row sums give A000108.
%Y A339754 Main diagonal gives A001405.
%Y A339754 Column k=2 gives 2*A005817(n-3) (for n>2).
%K A339754 nonn,tabl
%O A339754 1,3
%A A339754 _Sergi Elizalde_, Feb 12 2021