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A298646 a(n) is the sum of the degrees of asymmetry of all Dyck paths of semilength n. 3
0, 0, 4, 18, 88, 360, 1524, 6090, 24784, 98244, 393820, 1556324, 6196656, 24461424, 97079220, 383132250, 1518103840, 5992343940, 23726184372, 93686670220, 370840981680, 1464969055368, 5798679839524, 22917832613988, 90725318348448, 358737952183800 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

All terms are even.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1664

FORMULA

a(n) = Sum_{k=0..n-1} k*A298645(n,k).

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

a(n) ~ 4^n / sqrt(Pi*n) * (1 - 2/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 06 2018

EXAMPLE

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.

MAPLE

a:= proc(n) option remember; `if`(n<4, [0$3, 4][n+1], (

      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)))

    end:

seq(a(n), n=1..30);  # Alois P. Heinz, Feb 21 2018

CROSSREFS

Cf. A000108, A298645.

Sequence in context: A164045 A178577 A130524 * A199309 A083325 A050146

Adjacent sequences:  A298643 A298644 A298645 * A298647 A298648 A298649

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Feb 21 2018

STATUS

approved

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Last modified July 18 07:10 EDT 2019. Contains 325134 sequences. (Running on oeis4.)