A103903 Square array T(n,k) read by antidiagonals: coordination sequence for lattice D_n.

1, 1, 24, 1, 40, 144, 1, 60, 370, 456, 1, 84, 792, 1640, 1056, 1, 112, 1498, 4724, 4930, 2040, 1, 144, 2592, 11620, 18096, 11752, 3504, 1, 180, 4194, 25424, 55650, 52716, 24050, 5544, 1, 220, 6440, 50832, 149568, 195972, 127816, 44200, 8256, 1, 264

Offset

4,3

Links

Table of n, a(n) for n=4..50.

M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, Zeit. f. Kristallographie, 212 (1997), 253-256 and arXiv:cond-mat/9706122.

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.

Formula

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

Example

1,24,144,456,1056,2040,3504,5544,8256,11736,
1,40,370,1640,4930,11752,24050,44200,75010,119720,
1,60,792,4724,18096,52716,127816,271908,524640,938652,
1,84,1498,11620,55650,195972,559258,1371316,2999682,6003956,
1,112,2592,25424,149568,629808,2100832,5910288,14610560,32641008,
1,144,4194,50832,361602,1801872,6976866,22413456,62407170,155242640,

Mathematica

nmin = 4; nmax = 13; f[x_, n_] := ((1/2)*((-1+Sqrt[x])^(2n)+(1+Sqrt[x])^(2n))*(1-x)^n) / (-1+x)^(2n)-(2n*x*(1+x)^(n-2)) / (1-x)^n; t = Table[ CoefficientList[ Series[ f[x, n], {x, 0, nmax-nmin} ], x], {n, nmin, nmax} ]; Flatten[ Table[ t[[n-k+1, k]], {n, 1, nmax-nmin+1}, {k, 1, n} ] ] (* Jean-François Alcover, Jan 24 2012, after g.f. *)

Prog

(PARI) T(n, k)={polcoeff(sum(i=0, n, (binomial(2*n, 2*i) - 2*n*binomial(n-2, i-1))*x^i)/(1-x)^n + O(x*x^k), k)} \\ Andrew Howroyd, Jul 03 2018

Crossrefs

Rows include A007900, A008355, A008357, A008359, A008361, A008376, A008378.

Cf. A103884.

Sequence in context: A097013 A040598 A097190 * A280631 A040599 A076721

Adjacent sequences: A103900 A103901 A103902 * A103904 A103905 A103906

Keyword

nonn,tabl

Author

Ralf Stephan, Feb 21 2005

Status

approved