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A159333 Roman factorial of n. 2
-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; text; internal format)
OFFSET

-2,5

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|.

LINKS

G. C. Greubel, Table of n, a(n) for n = -2..100

Daniel E. Loeb, A generalization of the binomial coefficients, arXiv:math/9502218 [math.CO], 1995.

D. Loeb, and G.-C. Rota, Formal Power Series of Logarithmic Type, Advances Math. 75, 1-118, 1989.

S. Roman, The Logarithmic Binomial Formula, Amer. Math. Monthly 99, 641-648, 1992.

Eric W. Weisstein, Roman Factorial.

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.

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.

MATHEMATICA

Table[If[n < 0, (-1)^(-n - 1)/(-n - 1)!, n!], {n, -2, 30}] (* G. C. Greubel, Jun 25 2018 *)

PROG

(PARI) for(n=-2, 30, print1(if(n<0, (-1)^(-n - 1)/(-n - 1)!, n!), ", ")) \\ G. C. Greubel, Jun 25 2018

(MAGMA) [-1, 1] cat [Factorial(n): n in [0..30]]; // G. C. Greubel, Jun 25 2018

CROSSREFS

Cf. A000142.

Equals, for n=>-1, all right hand columns of triangle A165680. - Johannes W. Meijer, Oct 16 2009

Sequence in context: A124355 A133942 * A165233 A000142 A104150 A074166

Adjacent sequences:  A159330 A159331 A159332 * A159334 A159335 A159336

KEYWORD

easy,sign

AUTHOR

Jonathan Vos Post, Apr 10 2009

STATUS

approved

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Last modified October 19 17:55 EDT 2018. Contains 316376 sequences. (Running on oeis4.)