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%I #32 Dec 21 2020 07:16:56
%S 0,0,4,18,88,360,1524,6090,24784,98244,393820,1556324,6196656,
%T 24461424,97079220,383132250,1518103840,5992343940,23726184372,
%U 93686670220,370840981680,1464969055368,5798679839524,22917832613988,90725318348448,358737952183800
%N a(n) is the sum of the degrees of asymmetry of all Dyck paths of semilength n.
%C The degree of asymmetry of a Dyck path is defined in the following manner: we label the steps of a Dyck path of length 2n, from left to right, by 1,2,..., n-1, n, n, n-1, ..., 2,1. The degree of asymmetry is defined to be the number of pairs of identically labeled steps that are not at the same level. Example: the Dyck path uduudd has degree of asymmetry 2. Indeed, the labels are 123321 and the steps labeled 2 are at different levels and those labeled 3 are also at different levels.
%C All terms are even.
%H Alois P. Heinz, <a href="/A298646/b298646.txt">Table of n, a(n) for n = 1..1664</a>
%F a(n) = Sum_{k=0..n-1} k*A298645(n,k).
%F a(n) = (2*(n-1)*(34328*n^3 + 1024539*n^2 - 2260739*n - 3203910)*n*a(n-1) + 24*(n-1)*(63276*n^4 - 396683*n^3 + 460919*n^2 + 544393*n - 1067970)* a(n-2) - 32*(34328*n^5 + 784243*n^4 - 7831140*n^3 + 24334466*n^2 - 31463717*n + 15037140)*a(n-3) - 128*(n-3)*(n-4)*(2*n-7)*(38875*n^2 - 225739*n + 246552)*a(n-4))/((n+2)*(n+1)*n*(56039*n^2 - 121145*n - 130144)) for n>3, a(n) = 0 for n<3, a(3) = 4. - _Alois P. Heinz_, Feb 28 2018
%F a(n) ~ 4^n / sqrt(Pi*n) * (1 - 2/sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 06 2018
%e a(3) = 4. Indeed, showing the step levels, the 5 = A000108(3) Dyck paths of semilength 3 are 111111, 122221, 123321, 111221, 122111. The first 3 are symmetric (degree of asymmetry 0) and each of the last 2 has degree of asymmetry 2.
%p a:= proc(n) option remember; `if`(n<4, [0$3, 4][n+1], (
%p 2*(n-1)*(34328*n^3+1024539*n^2-2260739*n-3203910)*n*a(n-1)
%p +24*(n-1)*(63276*n^4-396683*n^3+460919*n^2+544393*n-1067970)*
%p a(n-2)-32*(34328*n^5+784243*n^4-7831140*n^3+24334466*n^2
%p -31463717*n+15037140)*a(n-3)-128*(n-3)*(n-4)*(2*n-7)*(38875*n^2
%p -225739*n+246552)*a(n-4))/((n+2)*(n+1)*n*(56039*n^2-121145*n-130144)))
%p end:
%p seq(a(n), n=1..30); # _Alois P. Heinz_, Feb 21 2018
%t 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, 1, z] b[x - 1, y + i, v + j], {i, l}], {j, l}]][{-1, 1}]]]];
%t g[n_] := g[n] = Sum[b[n, j, j], {j, 0, n}];
%t T[n_, k_] := Coefficient[g[n], z, k];
%t a[n_] := Sum[k T[n, k], {k, 0, n - 1}];
%t Array[a, 30] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ in A298645 *)
%Y Cf. A000108, A298645.
%K nonn
%O 1,3
%A _Emeric Deutsch_, Feb 21 2018
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