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A296666
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Table read by rows, the even rows of A296664, T(n, k) for n >= 0 and 0 <= k <= 2n.
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4
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1, 1, 2, 1, 2, 5, 6, 5, 2, 5, 14, 19, 20, 19, 14, 5, 14, 42, 62, 69, 70, 69, 62, 42, 14, 42, 132, 207, 242, 251, 252, 251, 242, 207, 132, 42, 132, 429, 704, 858, 912, 923, 924, 923, 912, 858, 704, 429, 132
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OFFSET
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0,3
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COMMENTS
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Let v be the characteristic function of 1 (A063524) and M(n) for n >= 0 the symmetric Toeplitz matrix generated by the initial segment of v, then row n is the main diagonal of M(2n)^(2n).
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LINKS
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FORMULA
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T(n, n) are the central binomial coefficients A000984(n).
T(n, k) = binomial(2*n, n) - binomial(2*n, n+k+1) for k=0..n.
T(n, k) = binomial(2*n, n) - binomial(2*n, k-n-1) for k=n+1..2*n and n>0.
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EXAMPLE
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0: [ 1]
1: [ 1, 2, 1]
2: [ 2, 5, 6, 5, 2]
3: [ 5, 14, 19, 20, 19, 14, 5]
4: [ 14, 42, 62, 69, 70, 69, 62, 42, 14]
5: [ 42, 132, 207, 242, 251, 252, 251, 242, 207, 132, 42]
6: [132, 429, 704, 858, 912, 923, 924, 923, 912, 858, 704, 429, 132]
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MAPLE
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v := n -> `if`(n=1, 1, 0);
B := n -> LinearAlgebra:-ToeplitzMatrix([seq(v(j), j=0..n)], symmetric):
seq(convert(ArrayTools:-Diagonal(B(2*n)^(2*n)), list), n=0..10);
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MATHEMATICA
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v[n_] := If[n == 1, 1, 0];
m[n_] := MatrixPower[ToeplitzMatrix[Table[v[k], {k, 0, n}]], n];
d[n_] := If[n == 0, {1}, Diagonal[m[2 n]]];
Table[d[n], {n, 0, 6}] // Flatten
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PROG
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(Sage)
def T(n, k):
if k > n:
b = binomial(2*n, k - n - 1)
else:
b = binomial(2*n, n + k + 1)
return binomial(2*n, n) - b
for n in (0..6):
print([T(n, k) for k in (0..2*n)])
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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