0,3

a(n-5)=number of necklaces with 5 black beads and n-5 white beads.

a(-5-n)=a(n) for all integers.

The g.f. is Z(C_5,x), the 5-variate cycle index polynomial for the cyclic group C_5, with substitution x[i]->1/(1-x^i), i=1,...,5. Therefore by Polya enumeration a(n) is the number of cyclically inequivalent 5-necklaces whose 5 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_5,x). W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Feb 15 2005.

B. Sturmfels, Algorithms in Invariant Theory, Springer, '93, p. 65.

Index entries for Molien series

Index entries for sequences related to necklaces

G.f.: (1+x^2+3*x^3+4*x^4+6*x^5+4*x^6+3*x^7+x^8+x^10)/(1-x)/(1-x^2)/(1-x^3)/(1- x^4)/(1-x^5)

a(n) = ceiling(C(n, 5)/n) [with a different offset].

G.f.: (1-3*x+5*x^2-3*x^3+x^4)/((1-x)^4*(1-x^5)) - Michael Somos, Dec 04, 2001

(1+x^2+3*x^3+4*x^4+6*x^5+4*x^6+3*x^7+x^8+x^10)/(1-x)/(1-x^2)/(1-x^3)/(1- x^4)/(1-x^5)

k = 5; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] - Robert A. Russell (russell(AT)post.harvard.edu), Sep 27 2004

(PARI) a(n)=ceil((n+4)*(n+3)*(n+2)*(n+1)/120)

Cf. A000031, A047996.

Sequence in context: A057524 A011795 A051170 this_sequence A036830 A014153 A001924

Adjacent sequences: A008643 A008644 A008645 this_sequence A008647 A008648 A008649

nonn,easy

njas