0,3

Let C_n be the root lattice generated as a monoid by {+-2*e_i: 1 <= i <= n; +-e_i +- e_j: 1 <= i not equal to j <= n}. Let P(C_n) be the polytope formed by the convex hull of this generating set. Then the rows of (the signless version of) this array are the f-vectors of a unimodular triangulation of P(C_n) [Ardila et al.]. See A086645 for the corresponding array of h-vectors for these type C_n polytopes. See A063007 for the array of f-vectors for type A_n polytopes and A108556 for the array of f-vectors associated with type D_n polytopes. [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. p. 37, eq.(1.96) and p. 4. eq.(1.10).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 795.

W. Lang, Row polynomials.

Index entries for sequences related to Chebyshev polynomials.

F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices [From Peter Bala (pbala(AT)talktalk.net), Jun 25 2009]

a(n,m)=0 if n<m, a(0,0)=1 else a(n,m)=((-1)^(n-m))*(2^(2*m-1))*binomial(n+m,2*m)*(2*n)/(n+m).

O.g.f.: (1 + z*(1 - 2*x))/((1 + z)^2 - 4*x*z) = 1 + (-1 + 2*x)*z + (1 - 8*x + 8*x^2)*z^2 + ... . [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]

[1];[ -1,2];[1,-8,8];[ -1,18,-48,32];[1,-32,160,-256,128];...

The T-polynomial for row n=3, [ -1,18,-48,32], is T(2*3,x)=-1*x + 18*x^2 -48*x^4 + 32*x^6.

Cf. A075733 (different signs and offset). A084930 (coefficients of odd indexed T-polynomials).

Cf. A053120 (coefficients of T-polynomials, with interspersed zeros).

A086645, A063007, A108556. [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]

Sequence in context: A015152 A021461 A075733 this_sequence A145901 A123516 A016446

Adjacent sequences: A127671 A127672 A127673 this_sequence A127675 A127676 A127677

sign,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Mar 07 2007