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

%I A103883 #22 May 25 2023 07:29:55
%S A103883 1,1,8,1,18,16,1,32,74,24,1,50,224,170,32,1,72,530,768,306,40,1,98,
%T A103883 1072,2562,1856,482,48,1,128,1946,6968,8130,3680,698,56,1,162,3264,
%U A103883 16394,28320,20082,6432,954,64,1,200,5154,34624,83442,85992,42130,10304,1250,72
%N A103883 Square array A(n,k) read by antidiagonals: coordination sequence for lattice B_n.
%H A103883 G. C. Greubel, <a href="/A103883/b103883.txt">Antidiagonals n = 2..50, flattened</a>
%H A103883 M. Baake and U. Grimm, <a href="https://arxiv.org/abs/cond-mat/9706122">Coordination sequences for root lattices and related graphs</a>, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997.
%H A103883 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 A103883 Joan Serra-Sagrista, <a href="http://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 A103883 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.
%F A103883 From _G. C. Greubel_, May 24 2023: (Start)
%F A103883 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).
%F A103883 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)
%e A103883 Array, A(n, k), begins:
%e A103883   1,   8,    16,     24,      32,       40,        48, ... A022144;
%e A103883   1,  18,    74,    170,     306,      482,       698, ... A022145;
%e A103883   1,  32,   224,    768,    1856,     3680,      6432, ... A022146;
%e A103883   1,  50,   530,   2562,    8130,    20082,     42130, ... A022147;
%e A103883   1,  72,  1072,   6968,   28320,    85992,    214864, ... A022148;
%e A103883   1,  98,  1946,  16394,   83442,   307314,    907018, ... A022149;
%e A103883   1, 128,  3264,  34624,  216448,   954880,   3301952, ... A022150;
%e A103883   1, 162,  5154,  67266,  507906,  2653346,  10666146, ... A022151;
%e A103883   1, 200,  7760, 122264, 1099040,  6728168,  31208560, ... A022152;
%e A103883   1, 242, 11242, 210474, 2224178, 15804866,  83999962, ... A022153;
%e A103883   1, 288, 15776, 346304, 4254912, 34792672, 210482016, ... A022154;
%e A103883   ...
%e A103883 Antidiagonals, T(n, k), begin as:
%e A103883   1;
%e A103883   1,   8;
%e A103883   1,  18,   16;
%e A103883   1,  32,   74,    24;
%e A103883   1,  50,  224,   170,    32;
%e A103883   1,  72,  530,   768,   306,    40;
%e A103883   1,  98, 1072,  2562,  1856,   482,   48;
%e A103883   1, 128, 1946,  6968,  8130,  3680,  698,  56;
%e A103883   1, 162, 3264, 16394, 28320, 20082, 6432, 954, 64;
%t A103883 offset = 2;
%t A103883 T[n_, k_] := SeriesCoefficient[Sum[(Binomial[2n + 1, 2i] - 2i Binomial[n, i]) x^i, {i, 0, n}]/(1 - x)^n, {x, 0, k}];
%t A103883 Table[T[n - k, k], {n, offset, 11}, {k, 0, n - offset}] // Flatten (* _Jean-François Alcover_, Feb 13 2019 *)
%o A103883 (Magma)
%o A103883 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]]) >;
%o A103883 [A103883(n,k): k in [0..n-2], n in [2..14]]; // _G. C. Greubel_, May 24 2023
%o A103883 (SageMath)
%o A103883 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))
%o A103883 flatten([[A103883(n,k) for k in range(n-1)] for n in range(2,15)]) # _G. C. Greubel_, May 24 2023
%Y A103883 Rows include A022144, A022145, A022146, A022147, A022148, A022149, A022150, A022151, A022152, A022153, A022154.
%Y A103883 Columns include A001105.
%Y A103883 Cf. A103881, A103884.
%K A103883 nonn,tabl
%O A103883 2,3
%A A103883 _Ralf Stephan_, Feb 20 2005