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A321414 Array read by antidiagonals: T(n,k) is the number of n element multisets of the 2k-th roots of unity with zero sum. 5

%I

%S 0,0,1,0,2,0,0,3,0,1,0,4,2,3,0,0,5,0,6,0,1,0,6,0,10,6,4,0,0,7,4,15,0,

%T 12,0,1,0,8,0,21,2,20,12,5,0,0,9,0,28,24,35,0,21,0,1,0,10,6,36,0,64,

%U 10,35,22,6,0,0,11,0,45,0,84,84,70,0,33,0,1

%N Array read by antidiagonals: T(n,k) is the number of n element multisets of the 2k-th roots of unity with zero sum.

%C Equivalently, the number of closed convex paths of length n whose steps are the 2k-th roots of unity up to translation. For even n, there will be k paths of zero area consisting of n/2 steps in one direction followed by n/2 steps in the opposite direction.

%H Andrew Howroyd, <a href="/A321414/b321414.txt">Table of n, a(n) for n = 1..465</a>

%F G.f. of column k = 2^r: 1/(1 - x^2)^k - 1.

%F G.f. of column k = 2^r*p^e: ((2/(1 - x^p) - 1)/(1 - x^2)^p)^(k/p) - 1 for odd prime p.

%e Array begins:

%e =========================================================

%e n\k| 1 2 3 4 5 6 7 8 9 10 11 12

%e ---|-----------------------------------------------------

%e 1 | 0 0 0 0 0 0 0 0 0 0 0 0 ...

%e 2 | 1 2 3 4 5 6 7 8 9 10 11 12 ...

%e 3 | 0 0 2 0 0 4 0 0 6 0 0 8 ...

%e 4 | 1 3 6 10 15 21 28 36 45 55 66 78 ...

%e 5 | 0 0 6 0 2 24 0 0 54 4 0 96 ...

%e 6 | 1 4 12 20 35 64 84 120 183 220 286 396 ...

%e 7 | 0 0 12 0 10 84 2 0 270 40 0 624 ...

%e 8 | 1 5 21 35 70 174 210 330 657 715 1001 1749 ...

%e 9 | 0 0 22 0 30 236 14 0 1028 220 0 3000 ...

%e 10 | 1 6 33 56 128 420 462 792 2097 2010 3003 6864 ...

%e 11 | 0 0 36 0 70 576 56 0 3312 880 2 11976 ...

%e 12 | 1 7 50 84 220 926 924 1716 6039 5085 8008 24216 ...

%e ...

%e T(5, 3) = 6 because there are 6 rotations of the following figure:

%e o---o

%e / \

%e o---o---o

%e .

%e T(6, 3) = 12 because there are 4 basic shapes illustrated below which with rotations and reflections give 3 + 2 + 1 + 6 = 12 convex paths.

%e o o---o o---o

%e / \ / \ \ \

%e o===o===o===o o o o o o o

%e / \ \ / \ \

%e o---o---o o---o o---o

%o (PARI) \\ only supports k with at most one odd prime factor.

%o T(n, k)={my(r=valuation(k, 2), p); polcoef(if(k>>r == 1, 1/(1-x^2)^k + O(x*x^n), if(isprimepower(k>>r, &p), ((2/(1 - x^p) - 1)/(1 - x^2 + O(x*x^n))^p)^(k/p), error("Cannot handle k=", k) )), n)}

%Y Main diagonal is A321415.

%Y Columns include A053090(n+3), A321416, A321417, A321419.

%Y Cf. A103306, A103314, A262181, A292355.

%K nonn,tabl

%O 1,5

%A _Andrew Howroyd_, Nov 08 2018

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Last modified April 21 03:57 EDT 2021. Contains 343145 sequences. (Running on oeis4.)