cp's OEIS Frontend

A330417 Coefficient of e(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where e is the basis of elementary symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.

%I A330417 #7 Dec 15 2019 21:58:03
%S A330417 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 A330417 -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 A330417 -16,-7,8,-7,10,-11,17,-21,-18,12,-8,1,-9,-16,19,9,-11
%N A330417 Coefficient of e(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where e is the basis of elementary symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.
%C A330417 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A330417 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 A330417 a(n) = (-1)^(A056239(n) - Omega(n)) * A056239(n) * (Omega(n) - 1)! / Product c_i! where c_i is the multiplicity of prime(i) in the prime factorization of n.
%t A330417 Table[If[n==1,0,With[{tot=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]},(-1)^(tot-PrimeOmega[n])*tot*(PrimeOmega[n]-1)!/(Times@@Factorial/@FactorInteger[n][[All,2]])]],{n,30}]
%Y A330417 The unsigned version (except with a(1) = 1) is A319225.
%Y A330417 The transition from p to e by Heinz numbers is A321752.
%Y A330417 The transition from p to h by Heinz numbers is A321754.
%Y A330417 Different orderings with and without signs and first terms are A115131, A210258, A263916, A319226, A330415.
%Y A330417 Cf. A000041, A000110, A000258, A000670, A005651, A008480, A048994, A056239, A124794, A318762, A319191.
%K A330417 sign
%O A330417 1,3
%A A330417 _Gus Wiseman_, Dec 14 2019