43 lines
1.7 KiB
Text
43 lines
1.7 KiB
Text
<!-- Left Factorial !-->
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'''Left factorials''', <big><big>!n</big></big>, may refer to either ''subfactorials'' or to ''factorial sums'';
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<br>the same notation can be confusingly seen used for the two different definitions.
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Sometimes, ''subfactorials'' (also known as ''derangements'') may use any of the notations:
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:::::::* <big><big> <b> <span style="font-family:serif">!''n''`</span> </b> </big></big>
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:::::::* <big><big> <b> <span style="font-family:serif">!''n''</span> </b> </big></big>
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:::::::* <big><big> <b> <span style="font-family:serif">''n''¡</span> </b> </big></big>
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(It may not be visually obvious, but the last example uses an upside-down exclamation mark.)
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<br>This Rosetta Code task will be using this formula for '''left factorial''':
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<big><big>
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::: <math> !n = \sum_{k=0}^{n-1} k! </math>
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</big></big>
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where
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<big><big>
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::: <math>!0 = 0</math>
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</big></big>
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;Task
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Display the left factorials for:
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* zero through ten (inclusive)
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* 20 through 110 (inclusive) by tens
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<br>
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Display the length (in decimal digits) of the left factorials for:
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* 1,000, 2,000 through 10,000 (inclusive), by thousands.
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;Also see
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* The OEIS entry: [http://oeis.org/A003422 A003422 left factorials]
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* The MathWorld entry: [http://mathworld.wolfram.com/LeftFactorial.html left factorial]
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* The MathWorld entry: [http://mathworld.wolfram.com/FactorialSums.html factorial sums]
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* The MathWorld entry: [http://mathworld.wolfram.com/Subfactorial.html subfactorial]
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;Related task:
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* [http://rosettacode.org/wiki/Permutations/Derangements permutations/derangements (subfactorials)]
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<br><br>
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