A330415 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.

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

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

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Formula

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.

Mathematica

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}]

Prog

(PARI)
C(sig)={my(S=Set(sig)); my(n=vecsum(sig)); if(n==0, 0, n*(#sig-1)!*(-1)^(#sig-1)/prod(k=1, #S, (#select(t->t==S[k], sig))!))}
a(n)={my(f=factor(n)); C(if(n==1, [], concat(vector(#f~, i, primepi(f[i, 1]) * vector(f[i, 2], j, 1)))))} \\ Andrew Howroyd, Oct 15 2025

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, A330417.

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

Sequence in context: A066328 A373956 A330417 * A319225 A333627 A304037

Adjacent sequences: A330412 A330413 A330414 * A330416 A330417 A330418

Keyword

sign

Author

Gus Wiseman, Dec 14 2019

Status

approved