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

1,5

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.

Links

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

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.

Example

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

Prog

(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)}

Crossrefs

Main diagonal is A321415.

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

Cf. A103306, A103314, A262181, A292355.

Sequence in context: A212163 A212195 A228926 * A268865 A024159 A029302

Adjacent sequences: A321411 A321412 A321413 * A321415 A321416 A321417

Keyword

nonn,tabl

Author

Andrew Howroyd, Nov 08 2018

Status

approved