A085990 Number of topological types of polygons with 2n different sides.

0, 3, 60, 2520, 181440, 19958400, 3113510400, 653837184000, 177843714048000, 60822550204416000, 25545471085854720000, 12926008369442488320000, 7755605021665492992000000, 5444434725209176080384000000, 4420880996869850977271808000000

Offset: 1

Author

Sergey L. Dolmatov, Almir Dzhumaev (aalma(AT)mail.ru), Aug 18 2003

Keywords

• nonn
• easy

Comments

a(n) equals (-1)^n times the coefficient of sqrt(1-x^2)*(arcsin x)^2 in int (arcsin x)^(2n-1) dx. - John M. Campbell, Jul 20 2011

For n >= 4, also the number of distinct adjacency matrices of the n-Moebius ladder. - Eric W. Weisstein, Mar 31 2017

Examples

			For example: if n=1 then no polygon exists with 2 different sides. If n=2 then the polygon has 4 different sides A, B, C, D. In this case 3 different types of such 4-angle exist: (A, B, C, D), (A, B, D, C), (A, C, B, D).
		

Links

• Eric Weisstein's World of Mathematics, Adjacency Matrix
• Eric Weisstein's World of Mathematics, Moebius Ladder

Crossrefs

Cf. A009445.

Programs

Maple

A085990:=n->`if`(n=1,0,(2*n-1)!/2): seq(A085990(n), n=1..20); # Wesley Ivan Hurt, Mar 31 2015

Mathematica

nn = 32; a = Log[1/(1 - x^2)^(1/4)] - x^2/4; Prepend[Select[Range[0, nn]! CoefficientList[Series[a, {x, 0, nn}], x], # > 0 &], 0] (* Geoffrey Critzer, Dec 10 2011 *)
Join[{0}, (2 Range[2, 20] - 1)!/2] (* Wesley Ivan Hurt, Mar 31 2015 *)

PARI

a(n)=(2*n-1)!\2 \\ Charles R Greathouse IV, Dec 10 2011

Formula

a(n) = (n-1)*(2*n-1)*(2*n-3)!

a(n) = (2n-1)!/2 = A009445(n)/2, for n>1. - Wesley Ivan Hurt, Mar 31 2015