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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.

%I A300996 #13 Apr 06 2018 16:51:18
%S A300996 0,0,0,1,6,28,122,516,2142,8776,35630,143757,577548,2313400,9246556,
%T A300996 36899291,147073062,585662952,2330527172,9268803248,36847836764,
%U A300996 146441131058,581850689938,2311451765318,9181309423676,36466002547328,144826879361752,575173217031049
%N 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.
%H A300996 Alois P. Heinz, <a href="/A300996/b300996.txt">Table of n, a(n) for n = 0..50</a>
%H A300996 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A300996 a(n) = Sum_{k = -floor((n-1)^2/4)..floor((n-1)^2/4)} k * A300953(n,k).
%e A300996 a(3) = (1/2) * (0 -2 +2 +2 +0) = 1:
%e A300996    .______.
%e A300996    |  /\  |______.______.______.
%e A300996    | /  \ | /\/\ | /\   |   /\ |______.
%e A300996    |/____\|/____\|/__\/\|/\/__\|/\/\/\|
%e A300996    |9 -9=0|5-7=-2|7 -5=2|7 -5=2|3 -3=0|
%Y A300996 Cf. A002620, A057571, A300953.
%K A300996 nonn
%O A300996 0,5
%A A300996 _Alois P. Heinz_, Mar 17 2018