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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A239140 A138553 A069016

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

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Last modified June 19 19:03 EDT 2019. Contains 324222 sequences. (Running on oeis4.)