A213616 Triangle read by rows, coefficients of the Bernoulli nabla polynomials BN_{n}(x) times A144845(n) in descending order of powers.
1, 2, -3, 6, -18, 13, 2, -9, 13, -6, 30, -180, 390, -360, 119, 6, -45, 130, -180, 119, -30, 42, -378, 1365, -2520, 2499, -1260, 253, 6, -63, 273, -630, 833, -630, 253, -42, 30, -360, 1820, -5040, 8330, -8400, 5060, -1680, 239, 10, -135, 780, -2520, 4998, -6300
Offset: 0
Examples
bn(0,x) = 1, bn(1,x) = 2*x - 3, bn(2,x) = 6*x^2 - 18*x + 13, bn(3,x) = 2*x^3 - 9*x^2 + 13*x - 6, bn(4,x) = 30*x^4 - 180*x^3 + 390*x^2 - 360*x + 119, bn(5,x) = 6*x^5 - 45*x^4 + 130*x^3 - 180*x^2 + 119*x - 30.
Links
- Peter Luschny, The Computation and Asymptotics of the Bernoulli numbers.
Programs
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Maple
seq(seq(coeff(denom(bernoulli(i, x))*bernoulli(i, x - 1), x, i - j), j=0..i), i=0..12);
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Mathematica
Table[If[i == 0, 1, 1/First[ FactorTerms[ BernoulliB[i, x]]]]*Table[ Coefficient[ Denominator[ BernoulliB[i, x]]*BernoulliB[i, x-1], x, i-j], {j, 0, i}], {i, 0, 12}] // Flatten (* Jean-François Alcover, Sep 27 2013, after Maple *)
Formula
T(n,k) = A144845(n)*[x^(n-k)]BN{n}(x).
Comments