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

1, 0, 5, 0, 15, 2, 35, 10, 70, 30, 128, 70, 220, 140, 360, 254, 565, 430, 855, 690, 1255, 1060, 1795, 1570, 2510, 2256, 3440, 3160, 4630, 4330, 6132, 5820, 8005, 7690, 10315, 10008, 13135, 12850, 16545, 16300, 20634, 20450, 25500, 25400, 31250, 31260

Offset

0,3

Comments

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,7,3,-8,3,7,-6,-4,4,1,-1)

Formula

G.f.: (1 - x^10)/((1 - x^2)^5 * (1 - x^5)^2).

G.f.: (1 - x + x^2 - x^3 + x^4)/((1 + x + x^2 + x^3 + x^4)*(1 - x)^6*(1 + x)^4).

Mathematica

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 *)

Prog

(PARI) Vec((1 - x^10)/((1 - x^2)^5 * (1 - x^5)^2) + O(x^50))

Crossrefs

Column k=5 of A321414.

Cf. A053090, A070190.

Sequence in context: A290867 A027635 A291218 * A226372 A367191 A093782

Adjacent sequences: A321413 A321414 A321415 * A321417 A321418 A321419

Keyword

nonn,easy

Author

Andrew Howroyd, Nov 09 2018

Status

approved