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A159333
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Roman factorial of n.
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1
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-1, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000
(list; graph; refs; listen; history; internal format)
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OFFSET
| -2,5
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COMMENTS
| The Roman factorial is named after its inventor Steve Roman. Knuth proved that for any integer n, a(n)*a(-n) = ((-1)^n)*|n|.
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REFERENCES
| Loeb, D. and Rota, G.-C. "Formal Power Series of Logarithmic Type." Advances Math. 75, 1-118, 1989.
Roman, S. "The Logarithmic Binomial Formula." Amer. Math. Monthly 99, 641-648, 1992.
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LINKS
| Daniel E. Loeb, A generalization of the binomial coefficients, Feb 9, 1995.
Eric W. Weisstein, Roman Factorial.
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FORMULA
| a(n) = n! = for a nonnegative integer. a(n) = ((-1)^(n+1))/(-n-1)! for negative integer n. There is a gamma function formula for nonintegral n.
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EXAMPLE
| For n < -2 we have noninteger values of a(n). a(-3) = 1/2. a(-4) = -1/6. a(-5) = 1/24. a(-6) = -1/120.
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CROSSREFS
| Cf. A000142.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)
Equals, for n=>-1, all right hand columns of triangle A165680.
(End)
Sequence in context: A124355 A133942 * A165233 A074166 A130641 A129655
Adjacent sequences: A159330 A159331 A159332 * A159334 A159335 A159336
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KEYWORD
| easy,sign
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 10 2009
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