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A336517
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T(n, k) = numerator([x^k] b(n, x)), where b(n, x) = 2^n*Sum_{k=0..n} binomial(n, k) * Bernoulli(k, 1/2) * x^(n-k). Triangle read by rows, for 0 <= k <= n.
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2
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1, 0, 2, -1, 0, 4, 0, -2, 0, 8, 7, 0, -8, 0, 16, 0, 14, 0, -80, 0, 32, -31, 0, 28, 0, -80, 0, 64, 0, -62, 0, 392, 0, -224, 0, 128, 127, 0, -496, 0, 1568, 0, -1792, 0, 256, 0, 762, 0, -992, 0, 9408, 0, -1536, 0, 512, -2555, 0, 1524, 0, -4960, 0, 6272, 0, -3840, 0, 1024
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OFFSET
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0,3
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COMMENTS
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Consider polynomials B_a(n, x) = a^n*Sum_{k=0..n} binomial(n, k)*Bernoulli(k, 1/a)*x^(n - k), with a != 0. They form an Appell sequence, the case a = 1 are the Bernoulli polynomials. T(n, k) are the numerators of the coefficients of the polynomials in the case a = 2.
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LINKS
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FORMULA
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EXAMPLE
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Rational polynomials start, coefficients of [numerators | denominators]:
[ [1], [ 1]]
[[0, 2], [ 1, 1]]
[[-1, 0, 4], [ 3, 1, 1]]
[[0, -2, 0, 8], [ 1, 1, 1, 1]]
[[7, 0, -8, 0, 16], [15, 1, 1, 1, 1]]
[[0, 14, 0, -80, 0, 32], [ 1, 3, 1, 3, 1, 1]]
[[-31, 0, 28, 0, -80, 0, 64], [21, 1, 1, 1, 1, 1, 1]]
[[0, -62, 0, 392, 0, -224, 0, 128], [ 1, 3, 1, 3, 1, 1, 1, 1]]
[[127, 0, -496, 0, 1568, 0, -1792, 0, 256], [15, 1, 3, 1, 3, 1, 3, 1, 1]]
[[0, 762, 0, -992, 0, 9408, 0, -1536, 0, 512], [ 1, 5, 1, 1, 1, 5, 1, 1, 1, 1]]
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MAPLE
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Bcp := n -> 2^n*add(binomial(n, k)*bernoulli(k, 1/2)*x^(n-k), k=0..n):
polycoeff := p -> seq(numer(coeff(p, x, k)), k = 0..degree(p, x)):
Trow := n -> polycoeff(Bcp(n)): seq(Trow(n), n=0..10);
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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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