A053090 -id:A053090 - cp's OEIS frontend

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

Andrew Howroyd, Nov 08 2018

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.

			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

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

Main diagonal is A321415.
Columns include A053090(n+3), A321416, A321417, A321419.
Cf. A103306, A103314, A262181, A292355.

\\ 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)}

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.

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

Original entry on oeis.org

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

Andrew Howroyd, Nov 09 2018

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.

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)

Column k=5 of A321414.

Cf. A053090, A070190.

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 *)

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

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).

A321417 Number of n element multisets of the 12th roots of unity with zero sum.

Original entry on oeis.org

1, 0, 6, 4, 21, 24, 64, 84, 174, 236, 420, 576, 926, 1260, 1896, 2540, 3639, 4800, 6618, 8592, 11499, 14700, 19200, 24204, 30972, 38544, 48480, 59620, 73884, 89892, 109960, 132480, 160221, 191308, 229038, 271248, 321809, 378264, 445128, 519608, 606954, 704016

Offset: 0

Andrew Howroyd, Nov 09 2018

Equivalently, the number of closed convex paths of length n whose steps are the 12th roots of unity up to translation. For even n, there will be 6 paths of zero area consisting of n/2 steps in one direction followed by n/2 steps in the opposite direction.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-6,-6, 6,13,-2,-18, -2,13,6,-6, -6,3,2,-1)

Column k=6 of A321414.

Cf. A053090, A198808.

Vec(((2/(1 - x^3) - 1)/(1 - x^2)^3)^2 + O(x^40))

G.f.: ((2/(1 - x^3) - 1)/(1 - x^2)^3)^2.

G.f.: (1 - x + x^2)^2/((1 + x + x^2)^2*(1 - x)^8*(1 + x)^4).

A156231 Sequence gives the Poincaré series [or Poincare series] of an ordinal Hodge algebra, or algebra with straightening law, for a ring that the braid group on four strands acts on. It is Cohen-Macaulay.

Original entry on oeis.org

1, 0, 6, 8, 24, 48, 106, 192, 369, 624, 1080, 1728, 2787, 4248, 6498, 9528, 13962, 19824, 28066, 38760, 53334, 71936, 96618, 127680, 167983, 218040, 281784, 360024, 458037, 577080, 724098, 900936, 1116636, 1373808, 1684038

Offset: 0

Stephen P. Humphries, Feb 06 2009

			For n=2 the dimension of the degree two part is 6.

Stephen P. Humphries, Action of some braid groups on Hodge algebras. Comm. Algebra 26 (1998), no. 4, pages 1233-1242. See Proposition 3.4

Stephen P. Humphries, Action of some braid groups on Hodge algebras Comm. Algebra 26 (1998), no. 4, pages 1233-1242. See Proposition 3.4 on page 1241.

A053090 is a similar Poincaré series [or Poincare series] for a ring on which the three strand braid groups acts.

G.f.: 1-(-4*x^20+8*x^19+6*x^18-12*x^17-11*x^16-2*x^15+25*x^14+10*x^13 -12*x^12) / ((1+x+x^2)^2*(1+x)^5*(1+x^2)*(1-x)^9) -(-14*x^11-15*x^10 +14*x^9+17*x^8+4*x^7-16*x^6-12*x^5+10*x^4+4*x^3-6*x^2) / ((1+x+x^2)^2*(1+x)^5*(1+x^2)*(1-x)^9).

cp's OEIS Frontend

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.

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

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A321417 Number of n element multisets of the 12th roots of unity with zero sum.

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PARI

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A156231 Sequence gives the Poincaré series [or Poincare series] of an ordinal Hodge algebra, or algebra with straightening law, for a ring that the braid group on four strands acts on. It is Cohen-Macaulay.

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