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%I #26 Sep 08 2022 08:46:14
%S 0,2,4,10,28,84,264,858,2860,9724,33592,117572,416024,1485800,5348880,
%T 19389690,70715340,259289580,955277400,3534526380,13128240840,
%U 48932534040,182965127280,686119227300,2579808294648,9723892802904,36734706144304,139067101832008,527495903500720
%N Number of rooted asymmetrical polyenoids of type U_n* having n edges.
%C Essentially the same as A068875 and A002420. See those entries for much more information.
%H Vincenzo Librandi, <a href="/A262543/b262543.txt">Table of n, a(n) for n = 1..1000</a>
%H S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="http://dx.doi.org/10.1021/ci00026a012">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
%H S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="/A002057/a002057.pdf">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
%F From _Gennady Eremin_, Feb 22 2021: (Start)
%F G.f.: 1 - 2*x - sqrt(1 - 4*x).
%F G.f.: x*(A(x) - 1), where A(x) is the g.f. of A068875. (End)
%e G.f. = 2*x^2 + 4*x^3 + 10*x^4 + 28*x^5 + 84*x^6 + 264*x^7 + 858*x^8 + ...
%p A262543List := proc(m) local A, P, n; A := [0, 2]; P := [2];
%p for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
%p A := [op(A), P[-1]] od; A end: A262543List(29); # _Peter Luschny_, Mar 24 2022
%t Table[If[n==1,0,2 CatalanNumber[n-1]], {n,1,26}] (* _Peter Luschny_, Feb 27 2017 *)
%t Join[{0}, Rest[CoefficientList[Series[2 (1 - Sqrt[1 - 4 x])/(2 x), {x, 0, 30}], x]]] (* _Vincenzo Librandi_, Feb 28 2017 *)
%o (Magma) [0] cat [2*Catalan(n-1): n in [2..40]]; // _Vincenzo Librandi_, Feb 28 2017
%o (PARI) a(n)=if(n>1, 2*binomial(2*n-2,n-1)/n, 0) \\ _Charles R Greathouse IV_, Feb 28 2017
%Y Cf. A068875, A002420.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_, Oct 06 2015
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