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

%I A103903 #26 Jul 08 2018 02:45:43
%S A103903 1,1,24,1,40,144,1,60,370,456,1,84,792,1640,1056,1,112,1498,4724,4930,
%T A103903 2040,1,144,2592,11620,18096,11752,3504,1,180,4194,25424,55650,52716,
%U A103903 24050,5544,1,220,6440,50832,149568,195972,127816,44200,8256,1,264
%N A103903 Square array T(n,k) read by antidiagonals: coordination sequence for lattice D_n.
%H A103903 M. Baake and U. Grimm, <a href="http://arXiv.org/abs/cond-mat/9706122">Coordination sequences for root lattices and related graphs</a>, Zeit. f. Kristallographie, 212 (1997), 253-256 and arXiv:cond-mat/9706122.
%H A103903 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%H A103903 Joan Serra-Sagrista, <a href="https://dx.doi.org/10.1016/S0020-0190(00)00119-8">Enumeration of lattice points in l_1 norm</a>, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
%F A103903 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.
%e A103903 1,24,144,456,1056,2040,3504,5544,8256,11736,
%e A103903 1,40,370,1640,4930,11752,24050,44200,75010,119720,
%e A103903 1,60,792,4724,18096,52716,127816,271908,524640,938652,
%e A103903 1,84,1498,11620,55650,195972,559258,1371316,2999682,6003956,
%e A103903 1,112,2592,25424,149568,629808,2100832,5910288,14610560,32641008,
%e A103903 1,144,4194,50832,361602,1801872,6976866,22413456,62407170,155242640,
%t A103903 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. *)
%o A103903 (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
%Y A103903 Rows include A007900, A008355, A008357, A008359, A008361, A008376, A008378.
%Y A103903 Cf. A103884.
%K A103903 nonn,tabl
%O A103903 4,3
%A A103903 _Ralf Stephan_, Feb 21 2005