A336454 -id:A336454 - cp's OEIS frontend

A336517 T(n, k) = numerator([x^k] b(n, x)), where b(n, x) = 2^nSum_{k=0..n} binomial(n, k) Bernoulli(k, 1/2) * x^(n-k). Triangle read by rows, for 0 <= k <= n.

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

Offset: 0

Peter Luschny, Jul 24 2020

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.

			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]]

Cf. A285865 (denominators), A336454 (polynomial denominator), A141459, A157779, A285866.

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);

Denominator(b(n, 1)) = A141459(n).
Numerator(b(n, -1)) = A285866(n).
Numerator(b(n, 0)) = A157779(n).

A335953 T(n, k) = numerator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)2^k Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 7, 0, -2, 0, 1, 0, 7, 0, -10, 0, 1, -31, 0, 7, 0, -5, 0, 1, 0, -31, 0, 49, 0, -7, 0, 1, 127, 0, -124, 0, 98, 0, -28, 0, 1, 0, 381, 0, -124, 0, 294, 0, -12, 0, 1, -2555, 0, 381, 0, -310, 0, 98, 0, -15, 0, 1

Offset: 0

Peter Luschny, Jul 25 2020

			[0]   1
[1]   0,   1
[2]  -1,   0,    1
[3]   0,  -1,    0,    1
[4]   7,   0,   -2,    0,  1
[5]   0,   7,    0,  -10,  0,   1
[6] -31,   0,    7,    0, -5,   0,   1
[7]   0, -31,    0,   49,  0,  -7,   0,   1
[8] 127,   0, -124,    0, 98,   0, -28,   0, 1
[9]   0, 381,    0, -124,  0, 294,   0, -12, 0, 1

Peter H. N. Luschny, An introduction to the Bernoulli function, arXiv:2009.06743 [math.HO], 2020.

Cf. A285865 (denominators), A336454 (polynomial denominator), A336517, A001896, A001897.

Bcn := n -> 2^n*bernoulli(n, 1/2):
Bcp := n -> add(binomial(n, k)*Bcn(k)*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(print(Trow(n)), n=0..9);

cp's OEIS Frontend

A336517 T(n, k) = numerator([x^k] b(n, x)), where b(n, x) = 2^nSum_{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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A335953 T(n, k) = numerator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)2^k Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.

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cp's OEIS Frontend

A336517 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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Author

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Comments

Examples

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Programs

Maple

Formula

A335953 T(n, k) = numerator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)*2^k* Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.

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Author

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Programs

Maple

A336517 T(n, k) = numerator([x^k] b(n, x)), where b(n, x) = 2^nSum_{k=0..n} binomial(n, k) Bernoulli(k, 1/2) * x^(n-k). Triangle read by rows, for 0 <= k <= n.

A335953 T(n, k) = numerator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)2^k Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.