A082641 Triangle T(n,k) (n >= 1, 1 <= k <= n) read by rows, where T(n,k) = number of basic invariants of degree k for the cyclic group of order and degree n.

1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 4, 4, 4, 1, 3, 6, 6, 2, 2, 1, 3, 8, 12, 12, 6, 6, 1, 4, 10, 18, 16, 8, 4, 4, 1, 4, 14, 26, 32, 18, 12, 6, 6, 1, 5, 16, 36, 48, 32, 12, 8, 4, 4, 1, 5, 20, 50, 82, 70, 50, 30, 20, 10, 10, 1, 6, 24, 64, 104, 84, 36, 20, 12, 8, 4, 4, 1, 6, 28, 84, 168, 180, 132, 84, 60, 36, 24, 12, 12, 1, 7, 32, 104, 216, 242, 162, 96, 42, 30, 18, 12, 6, 6

Offset

1,6

Comments

T(n,k) is also the number of multisets of k integers ranging from 1 to n, such that the sum of members of the multiset is congruent to 0 mod n, and no submultiset exists whose sum of members is congruent to 0 mod n. - Andrew Weimholt, Jan 31 2011

References

M. D. Neusel and L. Smith, Invariant Theory of Finite Groups, Amer. Math. Soc., 2002; see p. 208.

C. W. Strom, Complete systems of invariants of the cyclic groups of equal order and degree, Proc. Iowa Acad. Sci., 55 (1948), 287-290.

Links

Table of n, a(n) for n=1..105.

Finklea, Moore, Ponomarenko and Turner, Invariant Polynomials and Minimal Zero Sequences, Involve 1 (2008), no. 2, 159-165.

Bryson W. Finklea, Terri Moore, Vadim Ponomarenko and Zachary J. Turner, Invariant polynomials and minimal zero sequences, Involve, 1:2 (2008), pp. 159-165.

Vadim Ponomarenko, Table (Excel spread-sheet format)

Vadim Ponomarenko, Programs

Example

Triangle with row sums (A002956):
  Z_1:  1  ................................... 1
  Z_2:  1  1  ................................ 2
  Z_3:  1  1  2  ............................. 4
  Z_4:  1  2  2  2  .......................... 7
  Z_5:  1  2  4  4  4  ...................... 15
  Z_6:  1  3  6  6  2  2  ................... 20
  Z_7:  1  3  8 12 12  6  6  ................ 48
  Z_8:  1  4 10 18 16  8  4  4  ............. 65
  Z_9:  1  4 14 26 32 18 12  6  6  ......... 119
  Z_10: 1  5 16 36 48 32 12  8  4  4  ...... 166
  Z_11: 1  5 20 50 82 70 50 30 20 10 10  ... 348
  ...

Crossrefs

Row sums give A002956.

Sequence in context: A118821 A118824 A209402 * A334507 A342462 A239140

Adjacent sequences: A082638 A082639 A082640 * A082642 A082643 A082644

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 15 2003

Extensions

More terms from Vadim Ponomarenko (vadim123(AT)gmail.com), Jun 29 2004

Status

approved