A319191 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.

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

Offset

1,5

Comments

A refinement of Stirling numbers of the first kind.

Links

Table of n, a(n) for n=1..47.

Formula

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!).

Mathematica

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}]

Crossrefs

An unsigned version is A124795.

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

Sequence in context: A133031 A275732 A200594 * A124795 A368547 A346560

Adjacent sequences: A319188 A319189 A319190 * A319192 A319193 A319194

Keyword

sign

Author

Gus Wiseman, Sep 13 2018

Status

approved