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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, of an undirected n-cycle.
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, 8030}]
allocated for Gus WisemanCoefficient 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.
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, 10, 12, 11, 1, -7, -8, -7, -9, 12, -9, -8, -6, 13, 14, 14, 7, 7, -10, 15, 6, -4, 7, -9, 8, 16, -7, -8, -7, -10, -11, 17, -21, 18, -12, 8, -1, -9, 16, 19, 9, -11, 16
1,3
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Up to sign, a(n) is the number of acyclic spanning subgraphs whose component sizes are the prime indices of n, of an undirected n-cycle.
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.
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, 80}]
The unsigned version (except with a(1) = 1) is A319225.
The transition from p to e by Heinz numbers is A321752.
The transition from p to h by Heinz numbers is A321754.
Different orderings with and without signs and first terms are A115131, A210258, A263916, A319225, A319226.
Cf. A000041, A000110, A000258, A000670, A005651, A008480, A048994, A056239, A124794, A318762, A319191.
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Gus Wiseman, Dec 14 2019
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editing
allocated for Gus Wiseman
allocated
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