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

0, 1, 5, 23, 102, 443, 1898, 8054, 33932, 142163, 592962, 2464226, 10209620, 42190558, 173962532, 715908428, 2941192472, 12065310083, 49428043442, 202249741418, 826671597572, 3375609654698, 13771567556012, 56138319705908, 228669994187432, 930803778591278

Offset: 0

Author

Alois P. Heinz, May 29 2015

Keywords

• nonn
• easy

Comments

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.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
• Wikipedia, Average, Arithmetic mean
• Wikipedia, Lattice path

Crossrefs

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

Programs

Magma

A258431:= func< n | n eq 0 select 0 else (4^(n-1) + Factorial(2*n-1)/Factorial(n-1)^2)/2 >;
[A258431(n): n in [0..40]]; // G. C. Greubel, Mar 18 2023

Maple

a:= proc(n) option remember; `if`(n<3, [0, 1, 5][n+1],
       ((8*n-10)*a(n-1)-(16*n-24)*a(n-2))/(n-1))
    end:
seq(a(n), n=0..30);

Mathematica

a[0]=0; a[1]=1; a[2]=5;
a[n_]:= a[n]= (2*(4*n-5)*a[n-1] - 8*(2*n-3)*a[n-2])/(n-1);
Table[a[n], {n,0,30}] (* Jean-François Alcover, May 31 2018, from Maple *)

SageMath

def A258431(n): return 0 if (n==0) else (4^(n-1) + factorial(2*n-1)/factorial(n-1)^2)/2
[A258431(n) for n in range(41)] # G. C. Greubel, Mar 18 2023

Formula

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

a(n) = (2*(4*n-5)*a(n-1) - 8*(2*n-3)*a(n-2))/(n-1) for n>2, a(0)=0, a(1)=1, a(2)=5.

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

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

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