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A212357 Coefficients for the cycle index polynomial for the cyclic group C_n multiplied by n, n>=1, read as partition polynomial. 3
1, 1, 1, 2, 0, 1, 2, 0, 1, 0, 1, 4, 0, 0, 0, 0, 0, 1, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

The partitions are ordered like in Abramowitz-Stegun (for the reference see A036036, where also a link to a work by C. F. Hindenburg from 1779 is found where this order has been used).

The row lengths sequence is A000041. The number of nonzero entries in row nr. n is  A000005(n).

The cycle index (multivariate polynomial) for the cyclic group C_n, called Z(C_n), is (sum(phi(k)*x_k^{n/k} ,k divides n))/n, n>=1, with Euler's totient function phi(n)= A000010(n). See the Harary and Palmer reference. For the coefficients of Z(C_n) in different tabulations see also A054523 and A102190.

REFERENCES

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 36, (2.2.10).

LINKS

Table of n, a(n) for n=1..87.

Wolfdieter Lang, Cycle index Z(C_n), n=1..15.

FORMULA

The cycle index polynomial for the cyclic group C_n is Z(C_n) = (a(n,k)*x[1]^(e[k,1])*x[2]^(e[k,2])*...*x[n]^(e[k,n]))/n, n>=1, if the k-th partition of n in Abramowitz-Stegun order is 1^(e[k,1]) 2^(e[k,2]) ... n^(e[k,n]), where a part j with vanishing exponent e[k,j] has to be omitted. The n dependence of the exponents has been suppressed. See the comment above for the Z(C_n) formula and the link for these polynomials for n=1..15.

a(n,k) is the coefficient the term of n*Z(C_n) corresponding to the k-th partition of n in Abramowitz-Stegun order. a(n,k) = 0 if there is no such term in Z(C_n).

EXAMPLE

n\k  1 2 3 4 5 6 7 8 9 10 11 ...

1:   1

2:   1 1

3:   2 0 1

4:   2 0 1 0 1

5:   4 0 0 0 0 0 1

6:   2 0 0 2 0 0 1 0 0  0  1

...

See the link for rows n=1..8 and the Z(C_n) polynomials for n=1..15.

n=6: Z(C_6) = (2*x[6] + 2*x[3]^2 + 1*x[2]^3 + x[1]^6)/6, because the relevant partitions of 6 appear for k=1: 6, k=4: 3^2, k=7: 2^3 and k=11: 1^6

CROSSREFS

Cf. A000005, A000010, A054523, A102190.

Sequence in context: A115723 A238160 A178524 * A114525 A127672 A294168

Adjacent sequences:  A212354 A212355 A212356 * A212358 A212359 A212360

KEYWORD

nonn,tabf

AUTHOR

Wolfdieter Lang, Jun 04 2012

STATUS

approved

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Last modified June 21 16:00 EDT 2018. Contains 305623 sequences. (Running on oeis4.)