%I A063007
%S A063007 1,1,2,1,6,6,1,12,30,20,1,20,90,140,70,1,30,210,560,630,252,1,42,420,
%T A063007 1680,3150,2772,924,1,56,756,4200,11550,16632,12012,3432,1,72,1260,
%U A063007 9240,34650,72072,84084,51480,12870,1,90,1980,18480,90090,252252
%N A063007 Triangle: T(n,k) = C(n,k)*C(n+k,k) read by rows.
%C A063007 T(n,k) is the number of compatible k-sets of cluster variables in Fomin
and Zelevinsky's Cluster algebra of finite type B_n. Take a row of
this triangle regarded as a polynomial in x and rewrite as a polynomial
in y := x+1. The coefficients of the polynomial in y give a row of
triangle A008459 (squares of binomial coefficients). For example
x^2+6*x+6=y^2+4*y+1. - Paul Boddington (psb(AT)maths.warwick.ac.uk),
Mar 07 2003
%C A063007 T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps
E=(1,0), N=(0,1) and D=(1,1) (i.e. bilateral Schroeder paths), having
k N=(0,1) steps. E.g. T(2,0)=1 because we have DD; T(2,1)=6 because
we have NED, NDE, EDN, END, DEN and DNE; T(2,2)=6 because we have
NNEE, NENE, NEEN, EENN, ENEN and ENNE. - Emeric Deutsch (deutsch(AT)duke.poly.edu),
Apr 20 2004
%C A063007 Another version of [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, . . .] DELTA [0,
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . . ] = 1; 1, 0; 1, 2, 0; 1, 6, 6,
0; 1, 12, 30, 20, 0; . . ., where DELTA is the operator defined in
A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr) Apr 15 2005
%C A063007 Terms in row n are the coefficients of the Legendre polynomial P(n,2x+1)
with increasing powers of x.
%C A063007 Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008:
(Start)
%C A063007 Row n of this triangle is the f-vector of the simplicial complex dual
to an associahedron of type B_n (a cyclohedron)[Fomin & Reading,
p.60]. See A008459 for the corresponding h-vectors for associahedra
of type B_n and A001263 and A033282 respectively for the h-vectors
and f-vectors for associahedra of type A_n.
%C A063007 An alternative description of this triangle in terms of f-vectors is
as follows. Let A_n be the root lattice generated as a monoid by
{e_i - e_j: 0 <= i,j <= n+1}. Let P(A_n) be the polytope formed by
the convex hull of this generating set. Then the rows of this array
are the f-vectors of a unimodular triangulation of P(A_n) [Ardila
et al.]. A008459 is the corresponding array of h-vectors for these
type A_n polytopes. See A127674 (without the signs) for the array
of f-vectors for type C_n polytopes and A108556 for the array of
f-vectors associated with type D_n polytopes.
%C A063007 The S-transform on the ring of polynomials is the linear transformation
of polynomials that is defined on the basis monomials x^k by S(x^k)
= binomial(x,k) = x(x-1)...(x-k+1)/k!. Let P_n(x) denote the S-transform
of the n-th row polynomial of this array. In the notation of [Hetyei]
these are the Stirling polynomials of the type B associahedra. The
first few values are P_1(x) = 2*x + 1, P_2(x) = 3*x^2 + 3*x + 1 and
P_3(x) = (10*x^3 + 15*x^2 + 11*x + 3)/3. These polynomials have their
zeros on the vertical line Re x = -1/2 in the complex plane, that
is, the polynomials P_n(-x) satisfy a Riemann hypothesis. See A142995
for further details. The sequence of values P_n(k) for k = 0,1,2,
3, ... produces the n-th row of A108625. (End)
%D A063007 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 366.
%D A063007 S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer.
Math. Soc. 15 (2002) no. 2, 497-529
%D A063007 S. Fomin and A. Zelevinsky, Y-Systems and generalized associahedra, Ann.
of Math. (2) 158 (2003), no. 3, 977-1018.
%D A063007 R. A. Sulanke, Objects counted by the central Delannoy numbers. J. Integer
Seq. 6 (2003), no. 1, Article 03.1.5.
%D A063007 Gabor Hetyei. The Stirling polynomial of a simplicial complex. Discrete
and Computational Geometry 35, Number 3, March 2006, pp 437-455.
[From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]
%H A063007 T. D. Noe, <a href="b063007.txt">Rows n=0..100 of triangle, flattened</
a>
%H A063007 S. Fomin and A. Zelevinsky, <a href="http://arXiv.org/abs/math/0104151">
Cluster algebras I: Foundations</a>, J. Amer. Math. Soc. 15 (2002),
no. 2, 497-529.
%H A063007 S. Fomin and A. Zelevinsky, <a href="http://projecteuclid.org/Dienst/
Repository/1.0/Disseminate/euclid.annm/1080003770/body/pdf">Y-systems
and generalized associahedra</a>, Ann. of Math. (2) 158 (2003), no.
3, 977-1018.
%H A063007 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/
Sulanke/delannoy.html">Objects counted by the central Delannoy numbers.</
a>, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.
%H A063007 V. Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">
Recurrences and Legendre transform</a>
%H A063007 F. Chapoton, <a href="http://www.mat.univie.ac.at/~slc/opapers/s51chapoton.html">
Enumerative properties of generalized associahedra</a>
%H A063007 F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, <a href="http:/
/arxiv.org/abs/0809.5123">Root polytopes and growth series of root
lattices</a> [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]
%H A063007 S. Fomin, N. Reading, <a href="http://arxiv.org/abs/math.CO/0505518">
Root systems and generalized associahedra</a>, Lecture notes for
IAS/Park-City 2004. [From Peter Bala (pbala(AT)toucansurf.com), Oct
28 2008]
%H A063007 G. Hetyei, <a href="http://www.math.uncc.edu/preprint/2004/2004_11.pdf">
Face enumeration using generalized binomial coefficients</a>. This
is the draft version of Hetyei's paper referenced above. [From Peter
Bala (pbala(AT)toucansurf.com), Oct 28 2008]
%F A063007 T(n, k) = (n+k)!/(k!^2*(n-k)!) = T(n-1, k)*(n+k)/(n-k) = T(n, k-1)*(n+k)*(n-k+1)/
k^2 = T(n-1, k-1)*(n+k)*(n+k-1)/k^2.
%F A063007 G.f.=G(t, z)=1/sqrt(1-2z-4tz+z^2). Row generating polynomials=P_n(1+2z),
i.e. T(n, k)=[z^k]P_n(1+2z), where P_n are the Legendre polynomials.
- Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 20 2004
%F A063007 Sum_{k>=0} T(n, k)*A000172(k) = Sum_{k>=0} T(n, k)^2 = A005259(n). -
Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 08 2005 - Philippe
DELEHAM (kolotoko(AT)wanadoo.fr), Jun 08 2005
%e A063007 1; 1,2; 1,6,6; 1,12,30,20; 1,20,90,140,70; ...
%p A063007 with(orthopoly): seq([1,seq(coeff(expand(P(n,1+2*z)),z^k),k=1..n)],n=0..9);
%o A063007 (PARI) T(n,k)=local(t); if(n<0,0,t=(x+x^2)^n; for(k=1,n,t=t'); polcoeff(t,
k)/n!)
%Y A063007 Columns include A000012, A002378, A033487 on the left and A000984, A002457,
A002544 on the right. Main diagonal is A006480. Row sums are A001850.
%Y A063007 Cf. A008459.
%Y A063007 Cf. A104684
%Y A063007 A033282 (f-vectors type A associahedra), A108625, A080721 (f-vectors
type D associahedra) [From Peter Bala (pbala(AT)toucansurf.com),
Oct 28 2008]
%Y A063007 Sequence in context: A133314 A049019 A046651 this_sequence A089231 A052296
A019538
%Y A063007 Adjacent sequences: A063004 A063005 A063006 this_sequence A063008 A063009
A063010
%K A063007 nonn,tabl,nice,easy
%O A063007 0,3
%A A063007 Henry Bottomley (se16(AT)btinternet.com), Jul 02 2001
%E A063007 More terms from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 08 2005
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