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.

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

Author

Peter Luschny, Jul 24 2020

Keywords

• sign
• frac
• tabl

Comments

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.

Examples

			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]]
		

Crossrefs

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

Programs

Maple

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);

Formula

Denominator(b(n, 1)) = A141459(n).

Numerator(b(n, -1)) = A285866(n).

Numerator(b(n, 0)) = A157779(n).