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%I #22 May 25 2017 04:17:59
%S 1,8,56,384,2640,18304,128128,905216,6449664,46305280,334721024,
%T 2434334720,17801072640,130809692160,965500108800,7154863964160,
%U 53214300733440,397094950010880,2972195534929920,22308469918924800
%N G.f.: c(2*x)^4, where c(x) is the g.f. of A000108.
%C a(n) is also the number of paths in a binary tree of length 2n+3 between two vertices that are 3 steps apart. - David Koslicki, (koslicki(AT)math.psu.edu), Nov 02 2010
%H G. C. Greubel, <a href="/A101596/b101596.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = ((8*n+12)/(3*n+12))*((3*n+3)/(n+3))*2^n*C(n+1), where C(n) and the Catalan numbers of A000108.
%F Conjecture: (n+4)*a(n)-4*(3n+7)*a(n-1)+16*(2n+1)*a(n-2)=0. - _R. J. Mathar_, Dec 13 2011
%F From _Benedict W. J. Irwin_, Jul 12 2016: (Start)
%F G.f.: (1-sqrt(1-8*x)+4*x*(2*x-2+sqrt(1-8*x)))/(32*x^4).
%F E.g.f: E^(4*x)*(2*x*(4*x-3)*BesselI(0,4*x) + (3-4*x+ 8*x^2)* BesselI(1, 4*x))/(4*x^3). (End)
%F a(n) ~ 2^(3*n+5)*n^(-3/2)/sqrt(Pi). - _Ilya Gutkovskiy_, Jul 12 2016
%t CoefficientList[Series[(1-Sqrt[1-8z]+4z(-2+Sqrt[1-8z]+2z))/(32z^4), {z, 0, 20}],z] (* _Benedict W. J. Irwin_, Jul 12 2016 *)
%o (PARI) x='x+O('x^50); Vec((1-sqrt(1-8*x) + 4*x*(2*x-2+ sqrt(1-8*x)) )/(32*x^4)) \\ _G. C. Greubel_, May 24 2017
%Y Cf. A085687, A003645, A052701.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Dec 08 2004