cp's OEIS Frontend

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

%I A085990 #31 Feb 16 2025 08:32:50
%S A085990 0,3,60,2520,181440,19958400,3113510400,653837184000,177843714048000,
%T A085990 60822550204416000,25545471085854720000,12926008369442488320000,
%U A085990 7755605021665492992000000,5444434725209176080384000000,4420880996869850977271808000000
%N A085990 Number of topological types of polygons with 2n different sides.
%C A085990 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
%C A085990 For n >= 4, also the number of distinct adjacency matrices of the n-Moebius ladder. - _Eric W. Weisstein_, Mar 31 2017
%H A085990 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AdjacencyMatrix.html">Adjacency Matrix</a>
%H A085990 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusLadder.html">Moebius Ladder</a>
%F A085990 a(n) = (n-1)*(2*n-1)*(2*n-3)!
%F A085990 a(n) = (2n-1)!/2 = A009445(n)/2, for n>1. - _Wesley Ivan Hurt_, Mar 31 2015
%e A085990 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).
%p A085990 A085990:=n->`if`(n=1,0,(2*n-1)!/2): seq(A085990(n), n=1..20); # _Wesley Ivan Hurt_, Mar 31 2015
%t A085990 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 *)
%t A085990 Join[{0}, (2 Range[2, 20] - 1)!/2] (* _Wesley Ivan Hurt_, Mar 31 2015 *)
%o A085990 (PARI) a(n)=(2*n-1)!\2 \\ _Charles R Greathouse IV_, Dec 10 2011
%Y A085990 Cf. A009445.
%K A085990 nonn,easy
%O A085990 1,2
%A A085990 Sergey L. Dolmatov, Almir Dzhumaev (aalma(AT)mail.ru), Aug 18 2003