A298122 - OEIS

A298122

a(n) is the number of n nonintersecting arches above the x-axis that start and/or end with an arch length equal to one and have floor((n+2)/2) arches starting in odd numbered positions.

%I #26 Jan 14 2018 16:57:32

%S 1,1,2,5,11,34,90,300,875,3038,9408,33516,108108,392040,1302444,

%T 4785066,16256955,60324550,208579800,780088452,2735682092,10296854984,

%U 36532677272,138231751840,495241833996,1882201158264,6799413051200,25939319270000,94374970110000

%N a(n) is the number of n nonintersecting arches above the x-axis that start and/or end with an arch length equal to one and have floor((n+2)/2) arches starting in odd numbered positions.

%F a(1) = a(2) = 1, a(3) = 2; for n > 3, a(n) = 2*(C(n-1, floor((n-1)/2))*C(n-2, floor((n-2)/2)) - (C(n-1, floor((n-2)/2))*C(n-2, floor((n-3)/2)))) - (C(n-3, floor((n-1)/2))*C(n-2, floor((n-1)/2))/(floor((n-1)/2)+1)).

%e Example: For n = 4 the a(4) = 5 solutions are as follows. (The numbers under the arches represent arches starting in an odd-numbered position on the x-axis.)

%e /\ /\

%e //\\ /\ /\ //\\ /\

%e ///\\\/\, //\\ /\ /\, /\//\\/\, /\///\\\, /\/\//\\.

%e 1 3 7 1 5 7 1 3 7 1 3 5 1 3 5

%Y Cf. A001263, A005558.

%K nonn

%O 1,3

%A _Roger Ford_, Jan 12 2018