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.

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, -21, 7, -17, 0, 84, 0, -70, 0, 28, -8, 0, 153, 0, -252, 0, 126, 0, -36, 9, 155, 0, -765, 0, 630, 0, -210, 0, 45, -10, 0, -1705, 0, 2805, 0, -1386, 0, 330, 0, -55, 11

Offset

0,4

Comments

Can be seen as the Bernoulli counterpart of the Euler triangles A247453 and A109449.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11325 (rows 0 <= n <= 150, flattened)

Digital Library of Mathematical Functions, Lerch’s Transcendent.

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

Eric Weisstein's World of Mathematics, Lerch Transcendent.

Formula

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).

T(n, 0) = 2*(2^n - 1)*Bernoulli(n, 1) = n*Euler(n - 1, 1) = -A226158(n).

Main diagonal is (-1)^(n+1)*n = A181983(n).

Example

B*(8, z) = 1024*(Zeta(-7, (z+1)/2) - Zeta(-7, z/2))
         = -17 + 84*z^2 - 70*z^4 + 28*z^6 - 8*z^7.
Triangle starts:
[ 0] [  0]
[ 1] [  1]
[ 2] [  1,    -2]
[ 3] [  0,    -3,    3]
[ 4] [ -1,     0,    6,   -4]
[ 5] [  0,     5,    0,  -10,   5]
[ 6] [  3,     0,  -15,    0,  15,    -6]
[ 7] [  0,   -21,    0,   35,   0,   -21,    7]
[ 8] [-17,     0,   84,    0, -70,     0,   28,  -8]
[ 9] [  0,   153,    0, -252,   0,   126,    0, -36,  9]
[10] [155,     0, -765,    0, 630,     0, -210,   0, 45, -10]
[11] [  0, -1705,    0, 2805,   0, -1386,    0, 330,  0, -55, 11]

Mathematica

B[n_, x_] := (-2)^n (BernoulliB[n, x/2] - BernoulliB[n, (x + 1)/2]);
Prepend[Table[CoefficientList[B[n, x], x], {n, 1, 11}], 0] // Flatten

Prog

(SageMath)
def Bstar(n, x):
    return (-2)^n*(bernoulli_polynomial(x/2, n) - bernoulli_polynomial((x+1)/2, n))
print(flatten([expand(Bstar(n, x)).list() for n in (0..11)]))

Crossrefs

Row sums are (-1)^n*A226158(n). Alternating row sums are A239977(n).

Cf. A181983, A247453, A109449, (A053382/A053383) Bernoulli polynomials.

Sequence in context: A180013 A094067 A094112 * A326926 A347564 A105569

Adjacent sequences: A333300 A333301 A333302 * A333304 A333305 A333306

Keyword

sign,tabf

Author

Peter Luschny, May 07 2020

Status

approved