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.

0, 0, 0, 1, 6, 28, 122, 516, 2142, 8776, 35630, 143757, 577548, 2313400, 9246556, 36899291, 147073062, 585662952, 2330527172, 9268803248, 36847836764, 146441131058, 581850689938, 2311451765318, 9181309423676, 36466002547328, 144826879361752, 575173217031049

Offset: 0

Author

Alois P. Heinz, Mar 17 2018

Keywords

• nonn

Examples

			a(3) = (1/2) * (0 -2 +2 +2 +0) = 1:
   .______.
   |  /\  |______.______.______.
   | /  \ | /\/\ | /\   |   /\ |______.
   |/____\|/____\|/__\/\|/\/__\|/\/\/\|
   |9 -9=0|5-7=-2|7 -5=2|7 -5=2|3 -3=0|
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..50
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A002620, A057571, A300953.

Formula

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