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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
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified May 9 02:35 EDT 2021. Contains 343685 sequences. (Running on oeis4.)