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A300323 Number of Dyck paths of semilength n such that the area under the right half of the path equals the area under the left half of the path.

%I A300323 #18 May 31 2018 10:09:14
%S A300323 1,1,2,3,6,12,28,69,186,522,1536,4638,14408,45568,146884,479871,
%T A300323 1589516,5320854,18000198,61412376,211282386,731973720,2553168136,
%U A300323 8957554412,31604599044,112060048354,399227283950,1428315878002,5130964125124,18499652813682
%N A300323 Number of Dyck paths of semilength n such that the area under the right half of the path equals the area under the left half of the path.
%H A300323 Alois P. Heinz, <a href="/A300323/b300323.txt">Table of n, a(n) for n = 0..300</a>
%H A300323 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A300323 a(n) >= A001405(n) with equality only for n <= 4.
%F A300323 a(n) is odd <=> n in { A000225 }.
%e A300323               /\
%e A300323              /  \      /\/\
%e A300323 a(3) = 3:   /    \    /    \    /\/\/\ .
%e A300323 .
%e A300323 a(5) = 12 counts A001405(5) = 10 symmetric plus 2 non-symmetric Dyck paths:
%e A300323              /\  /\
%e A300323           /\/  \/  \  and its reversal.
%p A300323 b:= proc(x, y) option remember; expand(`if`(x=0, 1,
%p A300323       `if`(y<1,   0, b(x-1, y-1)*z^(2*y-1))+
%p A300323       `if`(x<y+2, 0, b(x-1, y+1)*z^(2*y+1))))
%p A300323     end:
%p A300323 a:= proc(n) option remember; add((p-> add(coeff(p, z, i)^2
%p A300323       , i=0..degree(p)))(b(n, n-2*j)), j=0..n/2)
%p A300323     end:
%p A300323 seq(a(n), n=0..32);
%t A300323 b[x_, y_] := b[x, y] = Expand[If[x == 0, 1, If[y < 1, 0, b[x - 1, y - 1] z^(2y - 1)] + If[x < y + 2, 0, b[x - 1, y + 1] z^(2y + 1)]]];
%t A300323 a[n_] := a[n] = Sum[Function[p, Sum[Coefficient[p, z, i]^2, {i, 0, Exponent[p, z]}]][b[n, n - 2j]], {j, 0, n/2}];
%t A300323 Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, May 31 2018, from Maple *)
%Y A300323 Column k=0 of A300322.
%Y A300323 Cf. A000108 (all Dyck paths), A000225, A001405 (symmetric Dyck paths), A129182, A239927, A298645.
%K A300323 nonn
%O A300323 0,3
%A A300323 _Alois P. Heinz_, Mar 02 2018