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 A258431 Sum over all peaks of Dyck paths of semilength n of the arithmetic mean of the x and y coordinates. 2

%I

%S 0,1,5,23,102,443,1898,8054,33932,142163,592962,2464226,10209620,

%T 42190558,173962532,715908428,2941192472,12065310083,49428043442,

%U 202249741418,826671597572,3375609654698,13771567556012,56138319705908,228669994187432,930803778591278

%N Sum over all peaks of Dyck paths of semilength n of the arithmetic mean of the x and y coordinates.

%C A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

%H Alois P. Heinz, <a href="/A258431/b258431.txt">Table of n, a(n) for n = 0..1000</a>

%H Paul Barry, <a href="https://arxiv.org/abs/1912.01124">A Note on Riordan Arrays with Catalan Halves</a>, arXiv:1912.01124 [math.CO], 2019.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Average#Arithmetic_mean">Average, Arithmetic mean</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>

%F G.f.: x*(sqrt(1-4*x)+1)/(2*sqrt(1-4*x)^3).

%F a(n) = ((8*n-10)*a(n-1)-(16*n-24)*a(n-2))/(n-1) for n>2, a(0)=0, a(1)=1, a(2)=5.

%F a(n) = (4^(n-1)+(2*n-1)!/(n-1)!^2)/2 for n>0, a(0)=0.

%F a(n) = (A000302(n-1) + A002457(n-1))/2 for n>0, a(0)=0.

%F a(n) = (1/2)*binomial(2*n,n)*( 1 + 2*(n-1)/(n+1) + 3*(n-1)*(n-2)/((n+1)*(n+2)) + 4*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + 5*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) + ...) for n >= 1. - _Peter Bala_, Feb 17 2022

%p a:= proc(n) option remember; `if`(n<3, [0, 1, 5][n+1],

%p ((8*n-10)*a(n-1)-(16*n-24)*a(n-2))/(n-1))

%p end:

%p seq(a(n), n=0..30);

%t a = 0; a = 1; a = 5;

%t a[n_] := a[n] = ((8*n - 10)*a[n - 1] - (16*n - 24)*a[n - 2])/(n - 1);

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 31 2018, from Maple *)

%Y Cf. A000302, A000346, A000531, A002457, A002697, A002802, A029887.

%K nonn,easy

%O 0,3

%A _Alois P. Heinz_, May 29 2015

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Last modified August 18 02:34 EDT 2022. Contains 356204 sequences. (Running on oeis4.)