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%I #36 Jul 18 2023 08:17:53
%S 1,1,1,1,2,1,1,11,11,1,1,44,102,44,1,1,157,802,802,157,1,1,530,5551,
%T 10876,5551,530,1,1,1731,35121,124427,124427,35121,1731,1,1,5528,
%U 208732,1265704,2201030,1265704,208732,5528,1
%N Type D Eulerian triangle.
%C Let n >= 2 and write the polynomial D(n,0)+D(n,1)*x+...+D(n,n)*x^n as a polynomial in y := x-1. Then the coefficient of y^r is the number of cells of dimension n-r in the cellular decomposition of a Euclidean space containing a root system of type D_n. If n >= 2 then the corresponding row sum is 2^(n-1)*n!, while Sum_{k=0..n} 2^k*D(n,k) is given by sequence A080254. [Row sum formula corrected by _Joshua Swanson_, Jul 12 2022]
%C The entries in row n (for n >= 2) are the components of the h-vector of the permutohedra of type D_n. See A145902 for the corresponding array of f-vectors for type D permutohedra. [_Peter Bala_, Oct 29 2008]
%D K. S. Brown, Buildings, Springer-Verlag, 1988
%D T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 11.
%H Jose Bastidas, <a href="/A066094/b066094.txt">Table of n, a(n) for n = 0..1274</a> (First 50 rows)
%H Anna Borowiec and Wojciech Mlotkowski, <a href="http://arxiv.org/abs/1509.03758">New Eulerian numbers of type D</a>, arXiv:1509.03758 [math.CO], 2015.
%H C. Chow, <a href="http://arXiv.org/abs/math.CO/0201140">On the Eulerian polynomials of type D</a>, arXiv:math/0201140 [math.CO], 2002.
%H S.-M. Ma, <a href="http://arxiv.org/abs/1205.6242">Polynomials with only real zeros and the Eulerian polynomials of type D</a>, arXiv preprint arXiv:1205.6242 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 23 2012
%F Let D(n, k) denote the (k+1)st entry in the (n+1)st row and let A(n, k), B(n, k) be triangles A008292 (The Eulerian triangle), A060187 respectively. Then D(n, k) = B(n, k)-2^(n-1)*n*A(n-2, k-1).
%F Chow gives complicated recurrences and generating functions.
%F E.g.f.: [(1-x)*exp(z*(1-x)) - z*x*(1-x)*exp(2*z*(1-x))]/(1 - x*exp(2*z*(1-x))) = 1 + x*z + (1 + 2*x + x^2)*z^2/2! + (1 + 11*x + 11*x^2 + x^3)*z^3/3! + ... . [_Peter Bala_, Oct 29 2008]
%e From _Peter Bala_, Oct 29 2008: (Start)
%e The triangle begins
%e n\k|..0....1....2....3....4....5
%e ================================
%e 0..|..1
%e 1..|..1....1
%e 2..|..1....2....1
%e 3..|..1...11...11....1
%e 4..|..1...44..102...44....1
%e 5..|..1..157..802..802..157....1
%e ...
%e (End)
%Y Cf. A008292, A060187, A080254.
%Y Cf. A145902. [_Peter Bala_, Oct 29 2008]
%K easy,nonn,tabl
%O 0,5
%A _Paul Boddington_, Mar 05 2003
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