A103883 Square array A(n,k) read by antidiagonals: coordination sequence for lattice B_n.

1, 1, 8, 1, 18, 16, 1, 32, 74, 24, 1, 50, 224, 170, 32, 1, 72, 530, 768, 306, 40, 1, 98, 1072, 2562, 1856, 482, 48, 1, 128, 1946, 6968, 8130, 3680, 698, 56, 1, 162, 3264, 16394, 28320, 20082, 6432, 954, 64, 1, 200, 5154, 34624, 83442, 85992, 42130, 10304, 1250, 72

Offset: 2

Author

Ralf Stephan, Feb 20 2005

Keywords

• nonn
• tabl

Examples

			Array, A(n, k), begins:
  1,   8,    16,     24,      32,       40,        48, ... A022144;
  1,  18,    74,    170,     306,      482,       698, ... A022145;
  1,  32,   224,    768,    1856,     3680,      6432, ... A022146;
  1,  50,   530,   2562,    8130,    20082,     42130, ... A022147;
  1,  72,  1072,   6968,   28320,    85992,    214864, ... A022148;
  1,  98,  1946,  16394,   83442,   307314,    907018, ... A022149;
  1, 128,  3264,  34624,  216448,   954880,   3301952, ... A022150;
  1, 162,  5154,  67266,  507906,  2653346,  10666146, ... A022151;
  1, 200,  7760, 122264, 1099040,  6728168,  31208560, ... A022152;
  1, 242, 11242, 210474, 2224178, 15804866,  83999962, ... A022153;
  1, 288, 15776, 346304, 4254912, 34792672, 210482016, ... A022154;
  ...
Antidiagonals, T(n, k), begin as:
  1;
  1,   8;
  1,  18,   16;
  1,  32,   74,    24;
  1,  50,  224,   170,    32;
  1,  72,  530,   768,   306,    40;
  1,  98, 1072,  2562,  1856,   482,   48;
  1, 128, 1946,  6968,  8130,  3680,  698,  56;
  1, 162, 3264, 16394, 28320, 20082, 6432, 954, 64;
		

Links

• G. C. Greubel, Antidiagonals n = 2..50, flattened
• M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997.
• J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
• Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Crossrefs

Rows include A022144, A022145, A022146, A022147, A022148, A022149, A022150, A022151, A022152, A022153, A022154.

Columns include A001105.

Cf. A103881, A103884.

Programs

Magma

A103883:= func< n,k | (&+[Binomial(n-j-1,n-k-1)*(Binomial(2*n-2*k+1,2*j) - 2*j*Binomial(n-k,j)) : j in [0..k]]) >;
[A103883(n,k): k in [0..n-2], n in [2..14]]; // G. C. Greubel, May 24 2023

Mathematica

offset = 2;
T[n_, k_] := SeriesCoefficient[Sum[(Binomial[2n + 1, 2i] - 2i Binomial[n, i]) x^i, {i, 0, n}]/(1 - x)^n, {x, 0, k}];
Table[T[n - k, k], {n, offset, 11}, {k, 0, n - offset}] // Flatten (* Jean-François Alcover, Feb 13 2019 *)

SageMath

def A103883(n,k): return sum(binomial(n-j-1,n-k-1)*(binomial(2*n-2*k+1,2*j) - 2*j*binomial(n-k,j)) for j in range(k+1))
flatten([[A103883(n,k) for k in range(n-1)] for n in range(2,15)]) # G. C. Greubel, May 24 2023

Formula

G.f. of n-th row: (Sum_{i=0..n} (C(2n+1, 2*i) - 2*i*C(n, i))*x^i)/(1-x)^n.

From G. C. Greubel, May 24 2023: (Start)

A(n, k) = Sum_{j=0..k} binomial(n+k-j-1, n-1)*(binomial(2*n+1, 2*j) - 2*j*binomial(n, j)) (array).

T(n, k) = Sum_{j=0..k} binomial(n-j-1, n-k-1)*(binomial(2*n-2*k+1, 2*j) - 2*j*binomial(n-k, j)) (antidiagonals). (End)