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%I #5 Sep 15 2018 15:48:46
%S 1,1,-1,1,2,-3,-6,1,3,8,24,-6,-120,-30,-20,1,720,15,-5040,20,90,144,
%T 40320,-10,40,-840,-15,-90,-362880,-120,3628800,1,-504,5760,-420,45,
%U -39916800,-45360,3360,40,479001600,630,-6227020800,504,210,403200,87178291200
%N 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.
%C A refinement of Stirling numbers of the first kind.
%F 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!).
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t numPermsOfType[ptn_]:=Total[ptn]!/Times@@ptn/Times@@Factorial/@Length/@Split[ptn];
%t Table[(-1)^(Total[primeMS[n]]-PrimeOmega[n])*numPermsOfType[primeMS[n]],{n,100}]
%Y An unsigned version is A124795.
%Y Cf. A000041, A000110, A000258, A005651, A008480, A048994, A056239, A124794, A215366, A318762, A319182.
%K sign
%O 1,5
%A _Gus Wiseman_, Sep 13 2018
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