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A321416 Number of n element multisets of the 10th roots of unity with zero sum.

%I A321416 #14 Feb 28 2020 04:08:22
%S A321416 1,0,5,0,15,2,35,10,70,30,128,70,220,140,360,254,565,430,855,690,1255,
%T A321416 1060,1795,1570,2510,2256,3440,3160,4630,4330,6132,5820,8005,7690,
%U A321416 10315,10008,13135,12850,16545,16300,20634,20450,25500,25400,31250,31260
%N A321416 Number of n element multisets of the 10th roots of unity with zero sum.
%C A321416 Equivalently, the number of closed convex paths of length n whose steps are the 10th roots of unity up to translation. For even n, there will be 5 paths of zero area consisting of n/2 steps in one direction followed by n/2 steps in the opposite direction.
%H A321416 Andrew Howroyd, <a href="/A321416/b321416.txt">Table of n, a(n) for n = 0..1000</a>
%H A321416 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4,-6,7,3,-8,3,7,-6,-4,4,1,-1)
%F A321416 G.f.: (1 - x^10)/((1 - x^2)^5 * (1 - x^5)^2).
%F A321416 G.f.: (1 - x + x^2 - x^3 + x^4)/((1 + x + x^2 + x^3 + x^4)*(1 - x)^6*(1 + x)^4).
%t A321416 LinearRecurrence[{1, 4, -4, -6, 7, 3, -8, 3, 7, -6, -4, 4, 1, -1},{1, 0, 5, 0, 15, 2, 35, 10, 70, 30, 128, 70, 220, 140}, 50] (* _Jinyuan Wang_, Feb 28 2020 *)
%o A321416 (PARI) Vec((1 - x^10)/((1 - x^2)^5 * (1 - x^5)^2) + O(x^50))
%Y A321416 Column k=5 of A321414.
%Y A321416 Cf. A053090, A070190.
%K A321416 nonn,easy
%O A321416 0,3
%A A321416 _Andrew Howroyd_, Nov 09 2018