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A228403 The number of boundary twigs for complete binary twigs. A twig is a vertex with one edge on the boundary and only one other descendant.

%I A228403 #29 Apr 29 2022 10:27:33
%S A228403 1,4,10,28,84,264,858,2860,9724,33592,117572,416024,1485800,5348880,
%T A228403 19389690,70715340,259289580,955277400,3534526380,13128240840,
%U A228403 48932534040,182965127280,686119227300,2579808294648,9723892802904,36734706144304,139067101832008,527495903500720,2004484433302736
%N A228403 The number of boundary twigs for complete binary twigs. A twig is a vertex with one edge on the boundary and only one other descendant.
%C A228403 Essentially twice the Catalan numbers (A000108).
%C A228403 A068875 without the 2. - _R. J. Mathar_, Aug 24 2013
%H A228403 G. C. Greubel, <a href="/A228403/b228403.txt">Table of n, a(n) for n = 1..1000</a>
%F A228403 G.f.: 2*x*C^2 - x where C is the g.f. for the Catalan numbers
%F A228403 a(n) ~ 2^(2*n+1)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 23 2013
%e A228403 For n  = 3 there are 5 complete binary trees. Of these UUUDUDDUDDUD consists of three twigs as does its mirror image UDUUDUUDUDDD. UUDUUDUDDDUD and UDUUUDUDDUDD each have one twig and UUDUDDUUDUDD has two.
%t A228403 Rest[CoefficientList[Series[(1-Sqrt[1-4*x]-2*x-x^2)/x,{x,0,20}],x]] (* _Vaclav Kotesovec_, Aug 23 2013 *)
%o A228403 (PARI)
%o A228403 x = 'x + O('x^66);
%o A228403 C = serreverse( x/( 1/(1-x) ) ) / x; \\ Catalan A000108
%o A228403 gf = 2*x*C^2 - x;
%o A228403 Vec(gf) \\ _Joerg Arndt_, Aug 22 2013
%o A228403 (Python)
%o A228403 from itertools import accumulate
%o A228403 def A228403_list(size):
%o A228403     if size < 1: return []
%o A228403     L, accu = [1], [2]
%o A228403     for _ in range(size-1):
%o A228403         accu = list(accumulate(accu + [accu[-1]]))
%o A228403         L.append(accu[-1])
%o A228403     return L
%o A228403 print(A228403_list(29)) # _Peter Luschny_, Apr 25 2016
%K A228403 nonn
%O A228403 1,2
%A A228403 _Louis Shapiro_, Aug 21 2013