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 A159333 Roman factorial of n. 2

%I

%S -1,1,1,1,2,6,24,120,720,5040,40320,362880,3628800,39916800,479001600,

%T 6227020800,87178291200,1307674368000,20922789888000,355687428096000,

%U 6402373705728000,121645100408832000,2432902008176640000

%N Roman factorial of n.

%C The Roman factorial is named after its inventor Steve Roman. Knuth proved that for any integer n, a(n)*a(-n) = ((-1)^n)*|n|.

%H G. C. Greubel, <a href="/A159333/b159333.txt">Table of n, a(n) for n = -2..100</a>

%H Daniel E. Loeb, <a href="http://arxiv.org/abs/math/9502218/">A generalization of the binomial coefficients</a>, arXiv:math/9502218 [math.CO], 1995.

%H D. Loeb, and G.-C. Rota, <a href="http://dx.doi.org/10.1016/0001-8708(89)90079-0">Formal Power Series of Logarithmic Type</a>, Advances Math. 75, 1-118, 1989.

%H S. Roman, <a href="http://www.jstor.org/stable/2324994">The Logarithmic Binomial Formula</a>, Amer. Math. Monthly 99, 641-648, 1992.

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/RomanFactorial.html">Roman Factorial.</a>

%F 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.

%e 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.

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

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

%o (MAGMA) [-1,1] cat [Factorial(n): n in [0..30]]; // _G. C. Greubel_, Jun 25 2018

%Y Cf. A000142.

%Y Equals, for n=>-1, all right hand columns of triangle A165680. - _Johannes W. Meijer_, Oct 16 2009

%K easy,sign

%O -2,5

%A _Jonathan Vos Post_, Apr 10 2009

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Last modified January 23 08:50 EST 2019. Contains 319376 sequences. (Running on oeis4.)