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

1,2

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

Links

Table of n, a(n) for n=1..15.

Eric Weisstein's World of Mathematics, Adjacency Matrix

Eric Weisstein's World of Mathematics, Moebius Ladder

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

Example

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).

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 *)

Prog

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

Crossrefs

Cf. A009445.

Sequence in context: A268964 A361536 A081854 * A202065 A036770 A201699

Adjacent sequences: A085987 A085988 A085989 * A085991 A085992 A085993

Keyword

nonn,easy

Author

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

Status

approved