%I #8 Dec 15 2019 21:57:55

%S 0,1,2,-1,3,-3,4,1,-2,-4,5,4,6,-5,-5,-1,7,5,8,5,-6,-6,9,-5,-3,-7,2,6,

%T 10,12,11,1,-7,-8,-7,-9,12,-9,-8,-6,13,14,14,7,7,-10,15,6,-4,7,-9,8,

%U 16,-7,-8,-7,-10,-11,17,-21,18,-12,8,-1,-9,16,19,9,-11,16

%N Coefficient of h(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where h is the basis of homogeneous symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%C Up to sign, a(n) is the number of acyclic spanning subgraphs of an undirected n-cycle whose component sizes are the prime indices of n.

%F a(n) = (-1)^(Omega(n) - 1) * A056239(n) * (Omega(n) - 1)! / Product c_i! where c_i is the multiplicity of prime(i) in the prime factorization of n.

%t Table[If[n==1,0,(-1)^(PrimeOmega[n]-1)*Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]*(PrimeOmega[n]-1)!/(Times@@Factorial/@FactorInteger[n][[All,2]])],{n,30}]

%Y The unsigned version (except with a(1) = 1) is A319225.

%Y The transition from p to e by Heinz numbers is A321752.

%Y The transition from p to h by Heinz numbers is A321754.

%Y Different orderings with and without signs and first terms are A115131, A210258, A263916, A319226, A330417.

%Y Cf. A000041, A000110, A000258, A000670, A005651, A008480, A048994, A056239, A124794, A318762, A319191.

%K sign

%O 1,3

%A _Gus Wiseman_, Dec 14 2019