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A341445 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having degree of symmetry k (n >= 1, 1 <= k <= n).

%I A341445 #23 Feb 13 2021 05:01:18
%S A341445 1,0,2,2,0,3,2,6,0,6,8,8,16,0,10,16,32,24,40,0,20,52,84,108,60,90,0,
%T A341445 35,134,262,294,310,150,210,0,70,432,816,1008,880,816,336,448,0,126,
%U A341445 1248,2544,3192,3208,2460,2100,784,1008,0,252
%N A341445 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having degree of symmetry k (n >= 1, 1 <= k <= n).
%C A341445 The degree of symmetry of a 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 the vertical line through the midpoint of the path.
%H A341445 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 A341445 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 A341445 For n=4 there are 6 Dyck paths with degree of symmetry equal to 2: uuuddudd, uuduuddd, uududdud, uuddudud, uduududd, ududuudd.
%e A341445 Triangle begins:
%e A341445      1;
%e A341445      0,    2;
%e A341445      2,    0,    3;
%e A341445      2,    6,    0,    6;
%e A341445      8,    8,   16,    0,   10;
%e A341445     16,   32,   24,   40,    0,   20;
%e A341445     52,   84,  108,   60,   90,    0,  35;
%e A341445    134,  262,  294,  310,  150,  210,   0,   70;
%e A341445    432,  816, 1008,  880,  816,  336, 448,    0, 126;
%e A341445   1248, 2544, 3192, 3208, 2460, 2100, 784, 1008,   0, 252;
%e A341445   ...
%p A341445 b:= proc(x, y, v) option remember; expand(
%p A341445       `if`(min(y, v, x-max(y, v))<0, 0, `if`(x=0, 1, (l-> add(add(
%p A341445       `if`(y=v+(j-i)/2, z, 1)*b(x-1, y+i, v+j), i=l), j=l))([-1, 1]))))
%p A341445     end:
%p A341445 g:= proc(n) option remember; add(b(n, j$2), j=0..n) end:
%p A341445 T:= (n, k)-> coeff(g(n), z, k):
%p A341445 seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, Feb 12 2021
%t A341445 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 == v + (j - i)/2, z, 1]*b[x - 1, y + i, v + j], {i, l}], {j, l}]][{-1, 1}]]]];
%t A341445 g[n_] := g[n] = Sum[b[n, j, j], {j, 0, n}];
%t A341445 T[n_, k_] := Coefficient[g[n], z, k];
%t A341445 Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Feb 13 2021, after _Alois P. Heinz_ *)
%Y A341445 Equivalent to A298645 with rows reversed.
%Y A341445 Row sums give A000108.
%Y A341445 Main diagonal gives A001405.
%Y A341445 Column k=1 gives A298647 (for n>2).
%Y A341445 Second subdiagonal gives 2*A191522.
%K A341445 nonn,tabl
%O A341445 1,3
%A A341445 _Sergi Elizalde_, Feb 12 2021