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%I #28 Oct 05 2023 02:18:05
%S 1,1,4,14,56,237,1046,4762,22198,105430,508384,2482297,12248416,
%T 60980875,305955356,1545397464,7852100294,40105277640,205798130604,
%U 1060467961508,5485199090812,28469067353686,148220323891460
%N Leftmost column sequence of triangle A073153.
%H Seiichi Manyama, <a href="/A073155/b073155.txt">Table of n, a(n) for n = 0..1000</a>
%H N. S. S. Gu, N. Y. Li and T. Mansour, <a href="http://dx.doi.org/10.1016/j.disc.2007.04.007">2-Binary trees: bijections and related issues</a>, Discr. Math., 308 (2008), 1209-1221.
%F Convolution of sequence formed from sum of adjacent terms yields the original sequence without the first term:
%F a(n+1) = Sum_{k=0..n} [a(k) + a(k-1)] * [a(n-k) + a(n-k-1)], where a(-1)=0.
%F G.f.: 1/2*(1-(1-4*x*(1+x)^2)^(1/2))/x/(1+x)^2. - _Vladeta Jovovic_, Oct 10 2003
%F a(n) = Sum_{k=0..n} C(2k,n-k)*C(k). - _Paul Barry_, Jul 09 2006
%F Conjecture: (n+1)*a(n) + (-3*n+4)*a(n-1) + 2*(-6*n+7)*a(n-2) + 2*(-6*n+11)*a(n-3) + 2*(-2*n+5)*a(n-4)=0. - _R. J. Mathar_, Nov 26 2012
%F G.f. A(x) satisfies: A(x) = 1 + x * ((1 + x) * A(x))^2. - _Ilya Gutkovskiy_, Jul 10 2020
%e a(3)=a(0)*[a(2)+a(1)]+[a(1)+a(0)]*[a(1)+a(0)]+[a(2)+a(1)]*a(0) =1*[4+1] + [1+1]*[1+1] + [4+1]*1 = 5 + 2*2 + 5 = 14.
%Y Cf. A073153, A073156, A073157.
%Y Cf. A052709, A360076, A360082, A360083.
%K easy,nonn
%O 0,3
%A _Paul D. Hanna_, Jul 29 2002
%E More terms from _Vladeta Jovovic_, Oct 10 2003
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