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

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, 93, 93, 37, 9, 1, 1, 10, 46, 130, 256, 130, 46, 10, 1, 1, 11, 56, 176, 386, 386, 176, 56, 11, 1, 1, 12, 67, 232, 562, 1024, 562, 232, 67, 12, 1

Offset: 0

Author

N. J. A. Sloane, Jan 17 2012

Keywords

• nonn
• tabl

Examples

			Triangle begins:
                   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    93    93    37    9    1
  1    10    46    130    256    130    46    10    1
1     11    56    176    386    386    176    56    11    1
...
		

Links

• Muniru A Asiru, Rows n=0..100 of triangle, flattened
• J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. VII Coordination sequences, Proc. R. Soc. Lond. A 453 (1997), 2369-2389.
• Hidefumi Ohsugi, Akiyoshi Tsuchiya, The h*-polynomials of locally anti-blocking lattice polytopes and their gamma-positivity, arXiv:1906.04719 [math.CO], 2019.
• Charles M. Wang, Josephine Yu, Toric h-vectors and Chow Betti Numbers of Dual Hypersimplices, arXiv:1707.04581 [math.CO], 2017.

Crossrefs

The triangle for Z^n is A007318, A_n is A008459, D_n is A108558, D*_n is A008518.

T(2n,n) gives A000302.

Programs

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

Mathematica

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 *)

Formula

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