|
%I #44 Jul 20 2019 12:29:00
%S 1,1,1,1,4,1,1,5,5,1,1,6,16,6,1,1,7,22,22,7,1,1,8,29,64,29,8,1,1,9,37,
%T 93,93,37,9,1,1,10,46,130,256,130,46,10,1,1,11,56,176,386,386,176,56,
%U 11,1,1,12,67,232,562,1024,562,232,67,12,1
%N Triangle read by rows: coordinator triangle for lattice A*_n.
%H Muniru A Asiru, <a href="/A204621/b204621.txt">Rows n=0..100 of triangle, flattened</a>
%H J. H. Conway and N. J. A. Sloane, <a href="https://doi.org/10.1098/rspa.1997.0126">Low-dimensional lattices. VII Coordination sequences</a>, Proc. R. Soc. Lond. A 453 (1997), 2369-2389.
%H Hidefumi Ohsugi, Akiyoshi Tsuchiya, <a href="https://arxiv.org/abs/1906.04719">The h∗-polynomials of locally anti-blocking lattice polytopes and their gamma-positivity</a>, arXiv:1906.04719 [math.CO], 2019.
%H Charles M. Wang, Josephine Yu, <a href="https://arxiv.org/abs/1707.04581">Toric h-vectors and Chow Betti Numbers of Dual Hypersimplices</a>, arXiv:1707.04581 [math.CO], 2017.
%F T(n, k) = Sum_{i=0..min(k,n-k)} binomial(n+1,i). [Wang and Yu, Theorem 4.1] - _Eric M. Schmidt_, Dec 07 2017
%e Triangle begins:
%e 1
%e 1 1
%e 1 4 1
%e 1 5 5 1
%e 1 6 16 6 1
%e 1 7 22 22 7 1
%e 1 8 29 64 29 8 1
%e 1 9 37 93 93 37 9 1
%e 1 10 46 130 256 130 46 10 1
%e 1 11 56 176 386 386 176 56 11 1
%e ...
%t T[n_, k_] := Sum[Binomial[n+1, i] , {i, 0, Min[k, n-k]}]; Table[T[n,k], {n,0,10}, {k,0,n}] // Flatten (* _Amiram Eldar_, Dec 14 2018 *)
%o (GAP) Flat(List([0..10],n->List([0..n],k->Sum([0..Minimum(k,n-k)],i->Binomial(n+1,i))))); # _Muniru A Asiru_, Dec 14 2018
%Y The triangle for Z^n is A007318, A_n is A008459, D_n is A108558, D*_n is A008518.
%Y T(2n,n) gives A000302.
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_, Jan 17 2012
|
