A285739
Numerator of discriminant of n-th Bernoulli polynomial.
Original entry on oeis.org
1, 1, 1, 28, 343, 31, 29791, 178035712, 11651995228221, 1087618835548371875, 13429024118357421875, 143533445691269324970571729935778225264543312, 91376242719004834465589805254054451484345405903423332764620213, 25397841834482816377486479267527401525220329290217
Offset: 1
1, 1/3, 1/16, 28/3375, 343/559872, 31/1815156, 29791/80621568, 178035712/124556484375, 11651995228221/80000000000000, ...
The first few Bernoulli polynomials are
0 | 1;
1 | x - 1/2;
2 | x^2 - x + 1/6;
3 | x^3 - 3*x^2/2 + x/2;
4 | x^4 - 2*x^3 + x^2 - 1/30;
5 | x^5 - 5*x^4/2 + 5*x^3/3 - x/6, etc.
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Table[Numerator[Discriminant[BernoulliB[n, x], x]], {n, 1, 14}]
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a(n) = numerator(poldisc(bernpol(n))); \\ Michel Marcus, Mar 02 2023
A285740
Denominator of discriminant of n-th Bernoulli polynomial.
Original entry on oeis.org
1, 3, 16, 3375, 559872, 1815156, 80621568, 124556484375, 80000000000000, 11881340006899968, 1218719480020992, 3405780508865246682482292626953125, 1526226812966134209666905971200000000000000000, 18160335421875000000000000
Offset: 1
1, 1/3, 1/16, 28/3375, 343/559872, 31/1815156, 29791/80621568, 178035712/124556484375, 11651995228221/80000000000000, ...
The first few Bernoulli polynomials are
0 | 1;
1 | x - 1/2;
2 | x^2 - x + 1/6;
3 | x^3 - 3*x^2/2 + x/2;
4 | x^4 - 2*x^3 + x^2 - 1/30;
5 | x^5 - 5*x^4/2 + 5*x^3/3 - x/6, etc.
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Table[Denominator[Discriminant[BernoulliB[n, x], x]], {n, 1, 14}]
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a(n) = denominator(poldisc(bernpol(n))); \\ Michel Marcus, Mar 02 2023
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.
Original entry on oeis.org
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
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]
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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
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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)]))
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