Antreas P. Hatzipolakis xpolakis at otenet.gr
Thu Sep 28 23:42:42 CEST 2000

ID Number: A000295 (Formerly M3416 and N1382)
Sequence:  0,0,1,4,11,26,57,120,247,502,1013,2036,4083,8178,16369,
Name:      Eulerian numbers 2^n - n - 1. (Column 2 of Euler's triangle A008292.)
References D. P. Roselle, Permutations by number of rises and successions, Proc.
              Amer. Math. Soc., 20 (1968), 8-16.
        D. E. Knuth, The Art of Computer Programming. Addison-Wesley,
              Reading, MA, Vol. 3, p. 34.
        F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p.
        R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics.
              Addison-Wesley, Reading, MA, 1990.
        J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958,
              p. 215.
        P. J. Cameron, Sequences realized by oligomorphic permutation
              groups, J. Integ. Seqs. Vol. 3 (2000), #P00.1.5.

The sequence in (b) below:

A simplex A_1A_2....A_n+1 is given in an affine n dimensional space; a_i
is the |n-1| face opposite A_i, P is a point not in any a_i and P_i is the
intersection of A_iP and a_i.
(a) Show that there is a unique quadric Q(P) tangent to a_i at P_i. Let
C(P) be the center of Q(P). In the relationship between P and C one
point C corresponds to a point P. Let h_n denote the number of points
P corresponding to a point C.
(b) Show that h_n = 2^n - n - 1.
(c) If P is at infinity, show that the locus of C has in barycentric
coordinates with respect to A_1A_2...A_n+1 the equation Sigma sqrt(x_i) = 0.
(Nieuw Archief voor Wiskunde 28 (1980) 115, #562 by O. Bottema)


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