proposed

approved

editing

proposed

Coefficient of p(y) / A056239(n)! in Product_{i >= 1} (1 + x_i), where p is power-sum symmetric functions and y is the integer partition with Heinz number n.

allocated for Gus WisemanCoefficient of p(y) in Product_{i >= 1} (1 + x_i), where p is power-sum symmetric functions and y is the integer partition with Heinz number n.

1, 1, -1, 1, 2, -3, -6, 1, 3, 8, 24, -6, -120, -30, -20, 1, 720, 15, -5040, 20, 90, 144, 40320, -10, 40, -840, -15, -90, -362880, -120, 3628800, 1, -504, 5760, -420, 45, -39916800, -45360, 3360, 40, 479001600, 630, -6227020800, 504, 210, 403200, 87178291200

1,5

A refinement of Stirling numbers of the first kind.

If n = Product prime(x_i)^y_i is the prime factorization of n, then a(n) = (-1)^(Sum x_i * y_i - Sum y_i) (Sum x_i * y_i)! / (Product x_i^y_i * Product y_i!).

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

numPermsOfType[ptn_]:=Total[ptn]!/Times@@ptn/Times@@Factorial/@Length/@Split[ptn];

Table[(-1)^(Total[primeMS[n]]-PrimeOmega[n])*numPermsOfType[primeMS[n]], {n, 100}]

An unsigned version is A124795.

Cf. A000041, A000110, A000258, A005651, A008480, A048994, A056239, A124794, A215366, A318762, A319182.

allocated

sign

Gus Wiseman, Sep 13 2018

approved

editing

allocated for Gus Wiseman

allocated

approved