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A319861
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Triangle read by rows, 0 <= k <= n: T(n,k) is the numerator of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; denominator is A319862.
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4
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1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 3, 1, 1, 1, 5, 5, 5, 5, 1, 1, 3, 15, 5, 15, 3, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 1, 7, 7, 35, 7, 7, 1, 1, 1, 9, 9, 21, 63, 63, 21, 9, 9, 1, 1, 5, 45, 15, 105, 63, 105, 15, 45, 5, 1, 1, 11, 55, 165, 165, 231, 231, 165, 165, 55, 11, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,8
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COMMENTS
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In Computer-Aided Geometric Design, the affine combination Sum_{k=0..n} (T(n,k)/A319862(n,k))*P_k is the halfway point for the Bézier curve of degree n defined by the control points P_k, k = 0, 1, ..., n.
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LINKS
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FORMULA
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T(n, k) = numerator of binomial(n,k)/2^n.
T(n, k) = binomial(n,k)/A082907(n,k).
T(n, k)/A319862(n,k) = binomial(n,k)/2^n.
T(n, n-k) = T(n,k).
T(n, 0) = 1.
Sum_{k=0..n} 2*k*T(n,k)/A319862(n,k) = n.
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 1, 1;
1, 3, 3, 1;
1, 1, 3, 1, 1;
1, 5, 5, 5, 5, 1;
1, 3, 15, 5, 15, 3, 1;
1, 7, 21, 35, 35, 21, 7, 1;
1, 1, 7, 7, 35, 7, 7, 1, 1;
1, 9, 9, 21, 63, 63, 21, 9, 9, 1;
1, 5, 45, 15, 105, 63, 105, 15, 45, 5, 1;
...
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MAPLE
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a:=(n, k)->binomial(n, k)/gcd(binomial(n, k), 2^n): seq(seq(a(n, k), k=0..n), n=0..11); # Muniru A Asiru, Sep 30 2018
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MATHEMATICA
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T[n_, k_] = Binomial[n, k]/GCD[Binomial[n, k], 2^n];
tabl[nn_] = TableForm[Table[T[n, k], {n, 0, nn}, {k, 0, n}]];
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PROG
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(Maxima)
T(n, k) := binomial(n, k)/gcd(binomial(n, k), 2^n)$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$
(GAP) Flat(List([0..11], n->List([0..n], k->Binomial(n, k)/Gcd(Binomial(n, k), 2^n)))); # Muniru A Asiru, Sep 30 2018
(Sage) flatten([[numerator(binomial(n, k)/2^n) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 19 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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