# A000295

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,
32752,65519,131054,262125,524268,1048555,2097130,4194281,
8388584,16777191,33554406,67108837,134217700,268435427,
536870882
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.
151.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics.
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)

Antreas

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