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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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%I #18 Dec 21 2024 15:20:12

%S 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,

%T 64,0,-62,0,392,0,-224,0,128,127,0,-496,0,1568,0,-1792,0,256,0,762,0,

%U -992,0,9408,0,-1536,0,512,-2555,0,1524,0,-4960,0,6272,0,-3840,0,1024

%N 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.

%C 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.

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

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

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

%e Rational polynomials start, coefficients of [numerators | denominators]:

%e [ [1], [ 1]]

%e [[0, 2], [ 1, 1]]

%e [[-1, 0, 4], [ 3, 1, 1]]

%e [[0, -2, 0, 8], [ 1, 1, 1, 1]]

%e [[7, 0, -8, 0, 16], [15, 1, 1, 1, 1]]

%e [[0, 14, 0, -80, 0, 32], [ 1, 3, 1, 3, 1, 1]]

%e [[-31, 0, 28, 0, -80, 0, 64], [21, 1, 1, 1, 1, 1, 1]]

%e [[0, -62, 0, 392, 0, -224, 0, 128], [ 1, 3, 1, 3, 1, 1, 1, 1]]

%e [[127, 0, -496, 0, 1568, 0, -1792, 0, 256], [15, 1, 3, 1, 3, 1, 3, 1, 1]]

%e [[0, 762, 0, -992, 0, 9408, 0, -1536, 0, 512], [ 1, 5, 1, 1, 1, 5, 1, 1, 1, 1]]

%p Bcp := n -> 2^n*add(binomial(n,k)*bernoulli(k,1/2)*x^(n-k), k=0..n):

%p polycoeff := p -> seq(numer(coeff(p, x, k)), k = 0..degree(p, x)):

%p Trow := n -> polycoeff(Bcp(n)): seq(Trow(n), n=0..10);

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

%K sign,frac,tabl,changed

%O 0,3

%A _Peter Luschny_, Jul 24 2020