OFFSET
1,7
COMMENTS
The row polynomials P(n,x) := Sum_{k=0..p(n)-1} a(n,k)*x^k, n >= 1, appear in the numerator of the g.f. G(p(n),x) for the numbers N(p(n),m) of inequivalent m-bead necklaces of two colors with p(n) beads of one color and m-p(n) beads of the other color. Here p(n)=A000040(n) (prime numbers). Equivalently, N(p(n),m) counts inequivalent necklaces with p(n) beads which are labeled with nonnegative numbers, such that the sum of the labels is m. For a proof of this equivalent formulation see a comment in A032191. Inequivalence is meant with respect to the cyclic group C_p(n).
This necklace g.f. is G(p(n),x) = P(n,x)/((1-x^p(n))*(1-x)^(p(n)-1)), n >= 1. The row polynomials P(n,x) are defined above. This g.f. is Z(C_p(n),x), the two variable (x[1] and x[p(n)]) cycle index polynomial for the cyclic group of prime order p(n), with substitution x[1]->1/(1-x^1)and x[p(n)]->1/(1-x^p(n)). This follows by Polya enumeration if the above mentioned labeled necklace problem is solved.
The row length sequence for this array a(n,k) is A000040(n) (n-th prime number), [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...].
The rows of this signed array are symmetric: a(n,k) = a(n,p(n)-1-k), n >= 2, k = 0..(p(n)-1)/2. See the explicit formula below.
The formulas for a(n,k), given below, produces in fact integers.
LINKS
FORMULA
EXAMPLE
Triangle begins:
[1, -0];
[1, -1, 1];
[1, -3, 5, -3, 1];
[1, -5, 13, -17, 13, -5, 1];
[1, -9, 41,-109, 191, -229, 191, -109, 41, -9, 1];
...
n=3: G(p(3),x)=G(5,x)=(1-3*x+5*x^2-3*x^3+1*x^4)/((1-x^5)*(1-x)^4) generates the necklace sequence A008646.
A103718(3,m), m=0..3, is [17,-17,7,-1]. Therefore (17-17*p(n)+7*p(n)^2-1*p(n)^3 )/3! gives, for n>=1, the third column [ -3,-17,-109,...].
CROSSREFS
KEYWORD
sign,easy,tabf
AUTHOR
Wolfdieter Lang, Feb 24 2005
STATUS
approved