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.

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

Offset: 1

Author

Andrew Howroyd, Nov 08 2018

Keywords

• nonn
• tabl

Comments

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.

Examples

			Array begins:
  =========================================================
  n\k| 1  2  3  4   5   6   7    8    9   10   11    12
  ---|-----------------------------------------------------
   1 | 0  0  0  0   0   0   0    0    0    0    0     0 ...
   2 | 1  2  3  4   5   6   7    8    9   10   11    12 ...
   3 | 0  0  2  0   0   4   0    0    6    0    0     8 ...
   4 | 1  3  6 10  15  21  28   36   45   55   66    78 ...
   5 | 0  0  6  0   2  24   0    0   54    4    0    96 ...
   6 | 1  4 12 20  35  64  84  120  183  220  286   396 ...
   7 | 0  0 12  0  10  84   2    0  270   40    0   624 ...
   8 | 1  5 21 35  70 174 210  330  657  715 1001  1749 ...
   9 | 0  0 22  0  30 236  14    0 1028  220    0  3000 ...
  10 | 1  6 33 56 128 420 462  792 2097 2010 3003  6864 ...
  11 | 0  0 36  0  70 576  56    0 3312  880    2 11976 ...
  12 | 1  7 50 84 220 926 924 1716 6039 5085 8008 24216 ...
  ...
T(5, 3) = 6 because there are 6 rotations of the following figure:
       o---o
      /     \
     o---o---o
.
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.
                        o        o---o     o---o
                       / \      /     \     \   \
    o===o===o===o     o   o    o       o     o   o
                     /     \    \     /       \   \
                    o---o---o    o---o         o---o
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..465

Crossrefs

Main diagonal is A321415.

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

Cf. A103306, A103314, A262181, A292355.

Programs

PARI

\\ only supports k with at most one odd prime factor.
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)}

Formula

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

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.