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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
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 FORMULA G.f.: x*(sqrt(1-4*x)+1)/(2*sqrt(1-4*x)^3). 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. 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 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] = ((8*n - 10)*a[n - 1] - (16*n - 24)*a[n - 2])/(n - 1); Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 31 2018, from Maple *) CROSSREFS Cf. A000302, A000346, A000531, A002457, A002697, A002802, A029887. Sequence in context: A085350 A113443 A124999 * A120902 A054441 A289803 Adjacent sequences:  A258428 A258429 A258430 * A258432 A258433 A258434 KEYWORD nonn,easy AUTHOR Alois P. Heinz, May 29 2015 STATUS approved

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Last modified July 5 21:22 EDT 2022. Contains 355102 sequences. (Running on oeis4.)