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A322507 a(n) is the number of arches for semi-meanders with n top arches that have exactly two covering arches.

%I A322507 #12 Aug 02 2021 14:11:16
%S A322507 0,0,0,2,6,28,82,310,918,3246,9762,33586,102490,348220,1076238,
%T A322507 3635050,11358178,38261430,120676406,406137648,1291309912,4345989212,
%U A322507 13914717064,46856195890
%N A322507 a(n) is the number of arches for semi-meanders with n top arches that have exactly two covering arches.
%F A322507 a(n) = Sum_{k=3..floor((n+4)/2)} (A259689(n+2,k)*(k-2)) - Sum_{k=2..floor((n+2)/2)} (A259689(n,k)*k*(k-1)).
%e A322507 Example: semi-meanders  n=4, -- top arches with two covering arches
%e A322507                   /\                    /\
%e A322507        /\        //\\     /\           //\\
%e A322507 /\ /\ //\\,  /\ ///\\\,  //\\ /\ /\,  ///\\\ /\
%e A322507                   --                    --          a(4) = 2.
%Y A322507 Cf. A259689.
%K A322507 nonn,more
%O A322507 1,4
%A A322507 _Roger Ford_, Dec 12 2018
%E A322507 Offset changed to 1 and a(14)-a(24) from _Jinyuan Wang_, Aug 02 2021