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

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..50

Wikipedia, Counting lattice paths

Formula

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

Example

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|

Crossrefs

Cf. A002620, A057571, A300953.

Sequence in context: A026851 A380024 A267689 * A181337 A002693 A289779

Adjacent sequences: A300993 A300994 A300995 * A300997 A300998 A300999

Keyword

nonn

Author

Alois P. Heinz, Mar 17 2018

Status

approved