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A204621 Triangle read by rows: coordinator triangle for lattice A*_n.

%I A204621 #47 Apr 25 2025 20:40:14
%S A204621 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 A204621 93,93,37,9,1,1,10,46,130,256,130,46,10,1,1,11,56,176,386,386,176,56,
%U A204621 11,1,1,12,67,232,562,1024,562,232,67,12,1
%N A204621 Triangle read by rows: coordinator triangle for lattice A*_n.
%H A204621 Muniru A Asiru, <a href="/A204621/b204621.txt">Rows n=0..100 of triangle, flattened</a>
%H A204621 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 A204621 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 A204621 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 A204621 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 A204621 Triangle begins:
%e A204621                    1
%e A204621                 1    1
%e A204621               1    4    1
%e A204621             1    5    5    1
%e A204621           1    6    16    6    1
%e A204621         1    7    22    22    7    1
%e A204621       1    8    29    64    29    8    1
%e A204621     1    9    37    93    93    37    9    1
%e A204621   1    10    46    130    256    130    46    10    1
%e A204621 1     11    56    176    386    386    176    56    11    1
%e A204621 ...
%t A204621 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 A204621 (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 A204621 The triangle for Z^n is A007318, A_n is A008459, D_n is A108558, D*_n is A008518.
%Y A204621 T(2n,n) gives A000302.
%K A204621 nonn,tabl
%O A204621 0,5
%A A204621 _N. J. A. Sloane_, Jan 17 2012