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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.

%I A336517 #18 Dec 21 2024 15:20:12
%S A336517 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 A336517 64,0,-62,0,392,0,-224,0,128,127,0,-496,0,1568,0,-1792,0,256,0,762,0,
%U A336517 -992,0,9408,0,-1536,0,512,-2555,0,1524,0,-4960,0,6272,0,-3840,0,1024
%N 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.
%C A336517 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 A336517 Denominator(b(n, 1)) = A141459(n).
%F A336517 Numerator(b(n, -1)) = A285866(n).
%F A336517 Numerator(b(n, 0)) = A157779(n).
%e A336517 Rational polynomials start, coefficients of [numerators | denominators]:
%e A336517                                            [ [1], [ 1]]
%e A336517                                        [[0,   2], [ 1, 1]]
%e A336517                                    [[-1, 0,   4], [ 3, 1, 1]]
%e A336517                              [[0,    -2, 0,   8], [ 1, 1, 1, 1]]
%e A336517                           [[7, 0,    -8, 0,  16], [15, 1, 1, 1, 1]]
%e A336517                     [[0,   14, 0,   -80, 0,  32], [ 1, 3, 1, 3, 1, 1]]
%e A336517                [[-31, 0,   28, 0,   -80, 0,  64], [21, 1, 1, 1, 1, 1, 1]]
%e A336517            [[0,  -62, 0,  392, 0,  -224, 0, 128], [ 1, 3, 1, 3, 1, 1, 1, 1]]
%e A336517       [[127, 0, -496, 0, 1568, 0, -1792, 0, 256], [15, 1, 3, 1, 3, 1, 3, 1, 1]]
%e A336517    [[0, 762, 0, -992, 0, 9408, 0, -1536, 0, 512], [ 1, 5, 1, 1, 1, 5, 1, 1, 1, 1]]
%p A336517 Bcp := n -> 2^n*add(binomial(n,k)*bernoulli(k,1/2)*x^(n-k), k=0..n):
%p A336517 polycoeff := p -> seq(numer(coeff(p, x, k)), k = 0..degree(p, x)):
%p A336517 Trow := n -> polycoeff(Bcp(n)): seq(Trow(n), n=0..10);
%Y A336517 Cf. A285865 (denominators), A336454 (polynomial denominator), A141459, A157779, A285866.
%K A336517 sign,frac,tabl
%O A336517 0,3
%A A336517 _Peter Luschny_, Jul 24 2020