A321753 Sum of coefficients of elementary symmetric functions in the power sum symmetric function indexed by the integer partition with Heinz number n.

1, 1, -1, 1, 1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, -1, -1, -1, -1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, -1, -1, -1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, -1, -1, 1, -1, 1

Offset

1

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Links

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

Wikipedia, Symmetric polynomial

Formula

a(n) = 1 if n is the Heinz number of an integer partition with an even number of even parts, otherwise a(n) = -1.

Totally multiplicative with a(prime(k)) = (-1)^(1 + k). - Andrew Howroyd, Nov 22 2025

Example

The sum of coefficients of p(32) = -6*e(32) + 6*e(221) + 3*e(311) - 5*e(2111) + e(11111) is a(15) = -1.

Prog

(PARI) a(n)=my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); (-1)^((1+primepi(p))*e)) \\ Andrew Howroyd, Nov 22 2025

Crossrefs

Row sums of A321752.

A322014 gives indices of positive values.

Cf. A005651, A008480, A056239, A124794, A124795, A135278, A296150, A319193, A319225, A319226, A321742-A321765.

Sequence in context: A108784 A127252 A244513 * A376149 A020985 A034947

Adjacent sequences: A321750 A321751 A321752 * A321754 A321755 A321756

Keyword

sign,mult

Author

Gus Wiseman, Nov 20 2018

Status

approved