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

Offset: 1

Author

Gus Wiseman, Nov 20 2018

Keywords

• nonn
• easy
• mult

Comments

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

Examples

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

Links

• Wikipedia, Symmetric polynomial.

Crossrefs

Row sums of A321897.

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

Cf. A000142, A000720.

Programs

Mathematica

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

Formula

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