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A300996 Total sum of half the difference between the area above and the area below the path, measured within the smallest enclosing rectangle based on the x-axis, in all Dyck paths of semilength n. 2

%I

%S 0,0,0,1,6,28,122,516,2142,8776,35630,143757,577548,2313400,9246556,

%T 36899291,147073062,585662952,2330527172,9268803248,36847836764,

%U 146441131058,581850689938,2311451765318,9181309423676,36466002547328,144826879361752,575173217031049

%N Total sum of half the difference between the area above and the area below the path, measured within the smallest enclosing rectangle based on the x-axis, in all Dyck paths of semilength n.

%H Alois P. Heinz, <a href="/A300996/b300996.txt">Table of n, a(n) for n = 0..50</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>

%F a(n) = Sum_{k = -floor((n-1)^2/4)..floor((n-1)^2/4)} k * A300953(n,k).

%e a(3) = (1/2) * (0 -2 +2 +2 +0) = 1:

%e .______.

%e | /\ |______.______.______.

%e | / \ | /\/\ | /\ | /\ |______.

%e |/____\|/____\|/__\/\|/\/__\|/\/\/\|

%e |9 -9=0|5-7=-2|7 -5=2|7 -5=2|3 -3=0|

%Y Cf. A002620, A057571, A300953.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Mar 17 2018

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Last modified November 22 01:14 EST 2019. Contains 329383 sequences. (Running on oeis4.)