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A330269 The number of semi-meanders with n top arches and concentric arches within the starting arch or a starting arch with length one.

%I A330269 #14 Feb 15 2020 10:44:42
%S A330269 1,1,2,4,8,18,42,108,282,786,2192,6402,18600,55978,167256,514102,
%T A330269 1567976,4896164,15170630,47957260,150468678,480371736,1522649458,
%U A330269 4900568718,15665593150,50761432998,163431901126,532624603680,1725349278270,5650796083020,18401781369182
%N A330269 The number of semi-meanders with n top arches and concentric arches within the starting arch or a starting arch with length one.
%H A330269 Jean-François Alcover, <a href="/A330269/b330269.txt">Table of n, a(n) for n = 1..46</a>
%F A330269 a(1) = 1, for n >= 2, a(n) = Sum_{k=1..n-1} A000682(k).
%e A330269 For n = 5, a(5) = 8:
%e A330269         /\                                  /\
%e A330269        //\\                      /\        /  \          /\
%e A330269       ///\\\      /\  /\        /  \      /  /\\    /\  //\\
%e A330269    /\////\\\\, /\//\\//\\, /\/\//\/\\, /\//\//\\\, //\\///\\\,
%e A330269                               /\
%e A330269                  /\          //\\          starting arch
%e A330269     /\  /\      //\\  /\    ///\\\        (1) (2) (3) (4)
%e A330269    //\\//\\/\, ///\\\//\\, ////\\\\/\,     4 + 2 + 1 + 1 = 8.
%t A330269 A000136 = Import["https://oeis.org/A000136/b000136.txt", "Table"][[All, 2]];
%t A330269 a[n_] := If[n == 1, 1, Sum[A000136[[k]]/k, {k, 1, n - 1}]];
%t A330269 a /@ Range[46] (* _Jean-François Alcover_, Feb 13 2020 *)
%Y A330269 Cf. A000682.
%K A330269 nonn
%O A330269 1,3
%A A330269 _Roger Ford_, Dec 07 2019
%E A330269 More terms from _Jinyuan Wang_, Dec 08 2019