A330415 -id:A330415 - 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.

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

Offset: 1

Gus Wiseman, Dec 14 2019

sign

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 of an undirected n-cycle whose component sizes are the prime indices of n.

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, A319226, A330415.
Cf. A000041, A000110, A000258, A000670, A005651, A008480, A048994, A056239, A124794, A318762, A319191.

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}]

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.

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.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula