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A333303 T(n, k) = [x^k] (-2)^n*(B(n, x/2) - B(n, (x+1)/2)) where B(n, x) are the Bernoulli polynomials. Triangle read by rows, for 0 <= k <= n.

%I A333303 #22 Dec 16 2020 19:15:29
%S A333303 0,1,1,-2,0,-3,3,-1,0,6,-4,0,5,0,-10,5,3,0,-15,0,15,-6,0,-21,0,35,0,
%T A333303 -21,7,-17,0,84,0,-70,0,28,-8,0,153,0,-252,0,126,0,-36,9,155,0,-765,0,
%U A333303 630,0,-210,0,45,-10,0,-1705,0,2805,0,-1386,0,330,0,-55,11
%N A333303 T(n, k) = [x^k] (-2)^n*(B(n, x/2) - B(n, (x+1)/2)) where B(n, x) are the Bernoulli polynomials. Triangle read by rows, for 0 <= k <= n.
%C A333303 Can be seen as the Bernoulli counterpart of the Euler triangles A247453 and A109449.
%H A333303 Michael De Vlieger, <a href="/A333303/b333303.txt">Table of n, a(n) for n = 0..11325</a> (rows 0 <= n <= 150, flattened)
%H A333303 Digital Library of Mathematical Functions, <a href="https://dlmf.nist.gov/25.14">Lerch’s Transcendent.</a>
%H A333303 Peter H. N. Luschny, <a href="https://arxiv.org/abs/2009.06743">An introduction to the Bernoulli function</a>, arXiv:2009.06743 [math.HO], 2020.
%H A333303 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LerchTranscendent.html">Lerch Transcendent.</a>
%F A333303 Let B*(n, x) denote the alternating Bernoulli rational polynomial functions defined by Z*(s, x) = Phi(-1, s, x) and B*(s, x) = -s Z*(1 - s, x). Here Phi(z, s, x) is the Hurwitz-Lerch transcendent defined as an analytic continuation of Sum_{k>=0} z^k/(k+x)^s. Then T(n, k) = (-1)^n [x^k] 2 B*(n, x).
%F A333303 T(n, 0) = 2*(2^n - 1)*Bernoulli(n, 1) = n*Euler(n - 1, 1) = -A226158(n).
%F A333303 Main diagonal is (-1)^(n+1)*n = A181983(n).
%e A333303 B*(8, z) = 1024*(Zeta(-7, (z+1)/2) - Zeta(-7, z/2))
%e A333303          = -17 + 84*z^2 - 70*z^4 + 28*z^6 - 8*z^7.
%e A333303 Triangle starts:
%e A333303 [ 0] [  0]
%e A333303 [ 1] [  1]
%e A333303 [ 2] [  1,    -2]
%e A333303 [ 3] [  0,    -3,    3]
%e A333303 [ 4] [ -1,     0,    6,   -4]
%e A333303 [ 5] [  0,     5,    0,  -10,   5]
%e A333303 [ 6] [  3,     0,  -15,    0,  15,    -6]
%e A333303 [ 7] [  0,   -21,    0,   35,   0,   -21,    7]
%e A333303 [ 8] [-17,     0,   84,    0, -70,     0,   28,  -8]
%e A333303 [ 9] [  0,   153,    0, -252,   0,   126,    0, -36,  9]
%e A333303 [10] [155,     0, -765,    0, 630,     0, -210,   0, 45, -10]
%e A333303 [11] [  0, -1705,    0, 2805,   0, -1386,    0, 330,  0, -55, 11]
%t A333303 B[n_, x_] := (-2)^n (BernoulliB[n, x/2] - BernoulliB[n, (x + 1)/2]);
%t A333303 Prepend[Table[CoefficientList[B[n, x], x], {n, 1, 11}], 0] // Flatten
%o A333303 (SageMath)
%o A333303 def Bstar(n,x):
%o A333303     return (-2)^n*(bernoulli_polynomial(x/2,n) - bernoulli_polynomial((x+1)/2,n))
%o A333303 print(flatten([expand(Bstar(n, x)).list() for n in (0..11)]))
%Y A333303 Row sums are (-1)^n*A226158(n). Alternating row sums are A239977(n).
%Y A333303 Cf. A181983, A247453, A109449, (A053382/A053383) Bernoulli polynomials.
%K A333303 sign,tabf
%O A333303 0,4
%A A333303 _Peter Luschny_, May 07 2020