A321898 - OEIS

A321898

Sum of coefficients of power sums symmetric functions in h(y) * Product_i y_i! where h is homogeneous symmetric functions and y is the integer partition with Heinz number n.

1, 1, 2, 1, 6, 2, 24, 1, 4, 6, 120, 2, 720, 24, 12, 1, 5040, 4, 40320, 6, 48, 120, 362880, 2, 36, 720, 8, 24, 3628800, 12, 39916800, 1, 240, 5040, 144, 4, 479001600, 40320, 1440, 6, 6227020800, 48, 87178291200, 120, 24, 362880, 1307674368000, 2, 576, 36, 10080

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OFFSET

1,3

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..1000

Wikipedia, Symmetric polynomial.

FORMULA

Totally multiplicative with a(p) = primepi(p)! = A000142(A000720(p)). - Amiram Eldar, Sep 10 2023

EXAMPLE

The sum of coefficients of 12h(32) = 2p(32) + 3p(221) + 2p(311) + 4p(2111) + p(11111) is a(15) = 12.

MATHEMATICA

f[p_, e_] := (PrimePi[p]!)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 10 2023 *)

PROG

(PARI) a(n) = { my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); (primepi(p)!)^e) } \\ Andrew Howroyd, Nov 10 2025

CROSSREFS

Row sums of A321897.

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

Cf. A000142, A000720.

Sequence in context: A185330 A217955 A325703 * A284434 A306543 A243484

Adjacent sequences: A321895 A321896 A321897 * A321899 A321900 A321901

KEYWORD

nonn,easy,mult

AUTHOR

Gus Wiseman, Nov 20 2018

STATUS

approved