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 Factorial

 http://functions.wolfram.com/06.01.04.0004.01

 Input Form

 Residue[n!, {n, -k}] == (-1)^(k - 1)/(k - 1)! /; Element[k, Integers] && k > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List["n", "!"]], ",", RowBox[List["{", RowBox[List["n", ",", " ", RowBox[List["-", "k"]]]], "}"]]]], "]"]], " ", "\[Equal]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]], "!"]]]]], " ", "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "0"]]]]]]]]

 MathML Form

 res n ( n ! ) ( - k ) ( - 1 ) k - 1 ( k - 1 ) ! /; k + Condition Subscript res n n -1 k -1 k -1 k -1 -1 k SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Residue", "[", RowBox[List[RowBox[List["n_", "!"]], ",", RowBox[List["{", RowBox[List["n_", ",", RowBox[List["-", "k_"]]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]], "!"]]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2001-10-29