They've been lazy, that's all: they have assumed that symmetry applies across the board, overlooking the fact that the basic recursion which they have broken at the origin is implied by a global property of the Gamma function; while symmetry is not. What intrigues me is that their identity reduction mechanism seems to give results correct also for integer arguments, despite the fact that they presumably must invoke properties of Gamma and hypergeometric functions which fail at the singularities. I conjecture that any identity supported in this fashion must hold for all flavours of pseudo-binomial coefficients defined by approaching the limit from a consistent direction. An example would be Rich's one plus two halves-wedge variant, where the limit is approached along x = 2 y rather than y = 0 . Which just goes to show that you can have half your cake and eat it! Fred On 7/10/13, Bill Gosper <billgosper@gmail.com> wrote:

rwg>Here is an almost correct technique for printing number triangles, of which Julian must repeatedly remind me:

pt[n_Integer, from_Integer: 0] := TableForm[Table[If[EvenQ[i + j], "", Binomial[-1 + i, -Floor[n/4] + (1 + i + j)/2] /. 0 -> ""], {i, from, n}, {j, -Ceiling[3*n/4], n + 1}]] [...] Duh, for actual symmetrical triangles, In[516]:= Column[Table[Row[Table[Binomial[n,k],{k,0,n}]," "],{n,0,5}],Center] Out[516]= 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 --rwg I somehow missed the distressing answer that Maple has been bodysnatched by the same brain virus (or is it Toxoplasma?) that's corrupted Mma:

http://isc.carma.newcastle.edu.au/standardCalc accepts Maple input.

Standard lookup results for *Pi^binomial(-2,-6)* Best guess: Pi^(5)

Both leading CASs! Maybe they think it's just a matter of personal

preference, like whether toilet paper spools off the front or the back? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun