Sage seems to use the definition from Concrete Mathematics by Graham, Knuth and Patashnik:
binomial(r, k) = r*(r-1)*...*(r-k+1)/(k * (k-1) *...* 1) if k >= 0 and r any real number, and binomial(r, k) = 0 if k < 0 That gives e.g. sage: binomial(-4, 5) -56 Though Peter's suggestion seems to make sense from one point of view, so does the above I would guess. Concrete Mathematics is seen as something of a bible in large parts of computer science, and it's unfortunate to disagree with a basic definition in there. The book has a long chapter devoted to convoluted identities on binomial coefficients -- I don't know how many of them are left true by the proposed change. Best, Johan Peter Luschny writes: > Hi all, > > I have already reported the unsatisfactory state of the binomial > function as implemented in Sage on 'Ask Sage' and had opened > a request for enhancement with ticket #17123. > > Here I take up a suggestion of John Palmieri to discuss the > issue on this list. The request is: > > Extend binomial(n,k) by the reflection formula, i. e. set > for n>=1 and k>=1 > > binomial(-k, -n) = (-1)^(n+k) binomial(n-1, k-1). > > The reasons are: > > (1) The current behavior introduces incompatibility with > Maple and Mathematica. This does not only question Sage as an > alternative for the other M-systems; because of the ubiquity > and importance of the binomial function it becomes difficult > to recommended Sage for implementing sequences on the OEIS. > > The behaviour of Maple and Mathematica (which support the > reflection formula) seems to be the de facto standard. Many > mathematicians write their formulas in the OEIS without > explicitly mentioning when they assume the reflection formula. > And this extension was used by mathematicians prior to Maple > and Mathematica. (So it is certainly not a question of 'do we > have to mimic all the crap of other M-systems' like some > commentators seem to suggest.) > > (2) The extension appears natural as the extended binomial > has the same behaviour as: > > def limit_binomial(n,r): > return limit(gamma(n+x+1)/(gamma(r+x+1)*gamma(n+x-r+1)), x=0) > > for n in (-5..5): print [limit_binomial(n,k) for k in (-5..5)] > > Indeed I find it confusing that the current implementation > does not exhibit this behavior. > > (3) Also the basic representation of Pascal's triangle in terms > of the matrix exponential (see the Wikipedia reference below) > makes the reflection extension appear natural as it is an extension > from the matrix with 1, 2, 3, ... subdiagonal and zero everywhere > else to the matrix with .., -3, -2, -1, 0, 1, 2, 3, .. as subdiagonal. > > (4) For a comparison of different extensions run this: > > def _binomial(n,k, pascal = true, reflection = true, uppernegation = true): > def b(n,k): return factorial(n)/(factorial(k)*factorial(n-k)) > if pascal and n >= 0 and k >= 0 and k <= n: return b(n,k) > if reflection and n < 0 and k < 0 and k <= n: return > (-1)^(n-k)*b(-k-1,n-k) > if uppernegation and n < 0 and k >= 0: return (-1)^k*b(k-n-1,k) > return 0 > > R = [n for n in (-5..5)] > print "(1) Pascal's triangle, for the ultra-orthodox." > print "(cf. [BP])" > for n in R: print [_binomial(n,k,true,false,false) for k in R] > print "(2) Current behavior of Sage, also Magma, extends (1)." > for n in R: print [_binomial(n,k,true,false,true) for k in R] > print "(3) Suggested behavior." > print "Also Maple and Mathematica, extends (1) and (2)." > print "(cf. [RS], p.5071)" > for n in R: print [_binomial(n,k,true,true,true) for k in R] > print "(4) Extends (1) by reflection only." > print "(cf. [LMMS], p.637)" > for n in R: print [_binomial(n,k,true,true,false) for k in R] > > (5) References > > [BP] Blaise Pascal; Traité du triangle arithmétique; G. Desprez, 1665. > [LMMS] Ana Luzón, Donatella Merlini, Manuel A. Morón, Renzo Sprugnoli; > Identities induced by Riordan arrays; Linear Algebra and its Applications, > 436 (2012) 631-647 > [WP] Pascal's triangle > https://en.wikipedia.org/wiki/Pascal's_triangle#Extensions > [MJK] M. J. Kronenburg; The Binomial Coefficient for Negative Arguments; > arXiv:1105.3689v2 > [RS] Renzo Sprugnoli; Negation of binomial coefficients; Discrete > Mathematics 308 (2008) 5070–5077 > [PL] Peter Luschny; Extensions of the binomial; Blog post; > http://oeis.org/wiki/User:Peter_Luschny/ExtensionsOfTheBinomial > [trac] http://trac.sagemath.org/ticket/17123 > > I'd like to hear your opinions. > > Peter -- -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
