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A103883
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Square array A(n,k) read by antidiagonals: coordination sequence for lattice B_n.
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1
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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
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OFFSET
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2,3
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LINKS
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J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
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FORMULA
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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.
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)
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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]]) >;
(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))
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CROSSREFS
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Rows include A022144, A022145, A022146, A022147, A022148, A022149, A022150, A022151, A022152, A022153, A022154.
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KEYWORD
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AUTHOR
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STATUS
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approved
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