The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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^(e[k,1])*x^(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 + 2*x^2 + 1*x^3 + x^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: A238160 A178524 A321731 * A114525 A127672 A294168 Adjacent sequences:  A212354 A212355 A212356 * A212358 A212359 A212360 KEYWORD nonn,tabf AUTHOR Wolfdieter Lang, Jun 04 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 17 16:03 EDT 2021. Contains 343980 sequences. (Running on oeis4.)