cp's OEIS Frontend

A228404 The number of complete binary trees with bicolored twigs. A twig is a vertex with one child on the boundary and the other child having no descendants.

%I A228404 #24 May 16 2025 00:33:41
%S A228404 1,2,8,24,76,249,836,2860,9932,34918,124032,444448,1604664,5831765,
%T A228404 21316860,78319140,289064460,1071275370,3984871440,14872552560,
%U A228404 55678270440,209027020410,786750047304,2968257334104,11223268563896,42522737574604,161415556062656,613813414982656
%N A228404 The number of complete binary trees with bicolored twigs. A twig is a vertex with one child on the boundary and the other child having no descendants.
%H A228404 G. C. Greubel, <a href="/A228404/b228404.txt">Table of n, a(n) for n = 0..1000</a>
%H A228404 S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt">Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017).
%F A228404 G.f.: 1 - x + 2*x*C^2 + x*C^4 where C is the g.f. for the Catalan numbers A000108.
%F A228404 Conjecture: -5*(n+3)*(n-2)*a(n) +5*(-n^2-n+18)*a(n-1) +5*(-n^2-n+48)*a(n-2) +(-5*n^2+20029*n+720)*a(n-3) +(-5*n^2-104153*n+186654)*
%F A228404 a(n-4) +(-5*n^2 +130153*n -508806)*a(n-5) +13650*(2*n-11)*(n-7)*a(n-6) = 0. - _R. J. Mathar_, Aug 08 2015
%F A228404 From _G. C. Greubel_, May 03 2021: (Start)
%F A228404 a(n) = C(n+2) - 2*C(n+1) + 2*C(n) with a(0) = 1, a(1) = 2, and C(n) = A000108(n).
%F A228404 E.g.f.: (-x^2*(1+x) + 2*exp(2*x)*(x*(1+x)*BesselI(0, 2*x) - (1+x^2)*BesselI(1, 2*x)))/x^2. (End)
%e A228404 For n = 2 there are two complete binary trees. Both consist of two twigs so can be colored 4 ways each.
%t A228404 Table[If[n<2, n+1, CatalanNumber[n+2] -2*CatalanNumber[n+1] +2*CatalanNumber[n]], {n,0,30}] (* _G. C. Greubel_, May 03 2021 *)
%o A228404 (PARI)
%o A228404 x = 'x + O('x^66);
%o A228404 C = serreverse( x/( 1/(1-x) ) ) / x; \\ Catalan A000108
%o A228404 gf = 1 - x + 2*x*C^2 + x*C^4;
%o A228404 Vec(gf) \\ _Joerg Arndt_, Aug 22 2013
%o A228404 (Magma) [1,2] cat [Catalan(n+2) -2*Catalan(n+1) +2*Catalan(n): n in [2..30]]; // _G. C. Greubel_, May 03 2021
%o A228404 (Sage) [1,2]+[catalan_number(n+2) -2*catalan_number(n+1) +2*catalan_number(n) for n in (2..30)] # _G. C. Greubel_, May 03 2021
%Y A228404 Without the bicoloring A228403 is the result.
%Y A228404 Cf. A000108.
%K A228404 nonn
%O A228404 0,2
%A A228404 _Louis Shapiro_, Aug 21 2013