A262543 Number of rooted asymmetrical polyenoids of type U_n* having n edges.

0, 2, 4, 10, 28, 84, 264, 858, 2860, 9724, 33592, 117572, 416024, 1485800, 5348880, 19389690, 70715340, 259289580, 955277400, 3534526380, 13128240840, 48932534040, 182965127280, 686119227300, 2579808294648, 9723892802904, 36734706144304, 139067101832008, 527495903500720

Offset: 1

Author

N. J. A. Sloane, Oct 06 2015

Keywords

• nonn
• easy

Comments

Essentially the same as A068875 and A002420. See those entries for much more information.

Examples

			G.f. = 2*x^2 + 4*x^3 + 10*x^4 + 28*x^5 + 84*x^6 + 264*x^7 + 858*x^8 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
• S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

Crossrefs

Cf. A068875, A002420.

Programs

Magma

[0] cat [2*Catalan(n-1): n in [2..40]]; // Vincenzo Librandi, Feb 28 2017

Maple

A262543List := proc(m) local A, P, n; A := [0, 2]; P := [2];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A262543List(29); # Peter Luschny, Mar 24 2022

Mathematica

Table[If[n==1,0,2 CatalanNumber[n-1]], {n,1,26}] (* Peter Luschny, Feb 27 2017 *)
Join[{0}, Rest[CoefficientList[Series[2 (1 - Sqrt[1 - 4 x])/(2 x), {x, 0, 30}], x]]] (* Vincenzo Librandi, Feb 28 2017 *)

PARI

a(n)=if(n>1, 2*binomial(2*n-2,n-1)/n, 0) \\ Charles R Greathouse IV, Feb 28 2017

Formula

From Gennady Eremin, Feb 22 2021: (Start)

G.f.: 1 - 2*x - sqrt(1 - 4*x).

G.f.: x*(A(x) - 1), where A(x) is the g.f. of A068875. (End)