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

1,3

Comments

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.

Links

Table of n, a(n) for n=1..69.

Formula

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.

Mathematica

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}]

Crossrefs

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.

Sequence in context: A326619 A326567 A066328 * A330415 A319225 A333627

Adjacent sequences: A330414 A330415 A330416 * A330418 A330419 A330420

Keyword

sign

Author

Gus Wiseman, Dec 14 2019

Status

approved