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%I #26 Mar 08 2020 04:51:10
%S 1,3,12,49,204,864,3714,16170,71178,316303,1417248,6396273,29051856,
%T 132700725,609200640,2809373915,13008512040,60457182345,281919911460,
%U 1318666411635,6185356518660,29088241615910,137121834221346,647821223533044,3066862717614234,14546629647573969
%N Catafusenes (see reference for precise definition).
%C 3-fold convolution of A002212. - _Emeric Deutsch_, Mar 13 2004
%H Vincenzo Librandi, <a href="/A045890/b045890.txt">Table of n, a(n) for n = 0..1000</a>
%H S. J. Cyvin et al., <a href="http://dx.doi.org/10.1021/ci00021a026">Enumeration and classification of certain polygonal systems representing polycyclic conjugated hydrocarbons: annelated catafusenes</a>, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
%F G.f.: (1 - x - sqrt(1-6*x+5*x^2))^3/(8*x^3). - _Emeric Deutsch_, Mar 13 2004
%F a(n) = (3/n)*Sum_{j=1..n} binomial(n, j)*binomial(2j+2, j-1) for n >= 1. - _Emeric Deutsch_, Mar 25 2004
%F Recurrence: (n+2)*(n+3)*a(n) = 2*(n+2)*(3*n+4)*a(n-1) - 5*(n-2)*(n+3)*a(n-2). - _Vaclav Kotesovec_, Oct 08 2012
%F a(n) ~ 6*5^(n+1/2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 08 2012
%p a := n->(3/n)*sum(binomial(n,j)*binomial(2*j+2,j-1),j=1..n): 1,seq(a(n),n=1..22);
%t a[n_] := 3*(Hypergeometric2F1[5/2, 1-n, 5, -4] + (n-1)*Hypergeometric2F1[7/2, 2-n, 6, -4]); a[0]=1; Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Jun 13 2012, after _Emeric Deutsch_ *)
%t CoefficientList[Series[(1-x-Sqrt[1-6x+5x^2])^3/(8x^3),{x,0,30}],x] (* _Harvey P. Dale_, Feb 07 2015 *)
%o (PARI) x='x+O('x^66); Vec((1-x-sqrt(1-6*x+5*x^2))^3/(8*x^3)) \\ _Joerg Arndt_, May 04 2013
%Y Cf. A002212.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Emeric Deutsch_, Mar 13 2004
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