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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.

%I A258431 #32 Mar 19 2023 02:48:30
%S A258431 0,1,5,23,102,443,1898,8054,33932,142163,592962,2464226,10209620,
%T A258431 42190558,173962532,715908428,2941192472,12065310083,49428043442,
%U A258431 202249741418,826671597572,3375609654698,13771567556012,56138319705908,228669994187432,930803778591278
%N A258431 Sum over all peaks of Dyck paths of semilength n of the arithmetic mean of the x and y coordinates.
%C A258431 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 A258431 Alois P. Heinz, <a href="/A258431/b258431.txt">Table of n, a(n) for n = 0..1000</a>
%H A258431 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 A258431 Wikipedia, <a href="https://en.wikipedia.org/wiki/Average#Arithmetic_mean">Average, Arithmetic mean</a>
%H A258431 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%F A258431 G.f.: x*(1 + sqrt(1-4*x))/(2*sqrt(1-4*x)^3).
%F A258431 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.
%F A258431 a(n) = (4^(n-1) + (2*n-1)!/(n-1)!^2)/2 for n>0, a(0) = 0.
%F A258431 a(n) = (A000302(n-1) + A002457(n-1))/2 for n>0, a(0) = 0.
%F A258431 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 A258431 a:= proc(n) option remember; `if`(n<3, [0, 1, 5][n+1],
%p A258431        ((8*n-10)*a(n-1)-(16*n-24)*a(n-2))/(n-1))
%p A258431     end:
%p A258431 seq(a(n), n=0..30);
%t A258431 a[0]=0; a[1]=1; a[2]=5;
%t A258431 a[n_]:= a[n]= (2*(4*n-5)*a[n-1] - 8*(2*n-3)*a[n-2])/(n-1);
%t A258431 Table[a[n], {n,0,30}] (* _Jean-François Alcover_, May 31 2018, from Maple *)
%o A258431 (Magma)
%o A258431 A258431:= func< n | n eq 0 select 0 else (4^(n-1) + Factorial(2*n-1)/Factorial(n-1)^2)/2 >;
%o A258431 [A258431(n): n in [0..40]]; // _G. C. Greubel_, Mar 18 2023
%o A258431 (SageMath)
%o A258431 def A258431(n): return 0 if (n==0) else (4^(n-1) + factorial(2*n-1)/factorial(n-1)^2)/2
%o A258431 [A258431(n) for n in range(41)] # _G. C. Greubel_, Mar 18 2023
%Y A258431 Cf. A000302, A000346, A000531, A002457, A002697, A002802, A029887.
%K A258431 nonn,easy
%O A258431 0,3
%A A258431 _Alois P. Heinz_, May 29 2015