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A103883
Square array A(n,k) read by antidiagonals: coordination sequence for lattice B_n.
1
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
2,3
LINKS
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.
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)
EXAMPLE
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;
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Ralf Stephan, Feb 20 2005
STATUS
approved