(MAGMAMagma) [Denominator((1/(2*m-2*k+1))*&+[Binomial(m, 2*k-i)*Binomial(2*m-2*k+i, i)*BernoulliNumber(i): i in [0..2*k]]): k in [0..m], m in [0..10]]; // Bruno Berselli, Jan 21 2013
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The coefficients F(m,i) are dual to Faulhaber coefficients, because they are obtained from the inverse expression Sum((k*(k + 1))^(m), k=0..N-1) to Faulhaber's formula from Sum((k)^(2*m-1), k=0..N-1) and there holds the identity F(m+i-1,i)=(-1)^i Fe(-m,i), where Fe(-m,i)=A093558(-m,i)/A093559(-m,i) is a Faulhaber coefficient for the sums of even powers of the first N-1 integers (for details see the reference 1, link, from p. 19).
A. Askar Dzhumadil'daev, D. Damir Yeliussizov, <a href="http://cs.uwaterloo.ca/journals/JIS/VOL16/Yeliussizov/dzhuma6.html">Power sums of binomial coefficients</a>, Journal of Integer Sequences, 16 (2013), Article 13.1.6
sign,nonn,frac,tabl,easy
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Table[a[m, k], {m, 0, 10}, {k, 0, m}] // Flatten (* _Jean-François Alcover, _, Jan 18 2013 *)
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21, 57, 17, 21, 13, 33, 63, 7, 5, 63, 4849845;
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The coefficients F(m,i) are dual to Faulhaber coefficients, because they are obtained from the inverse expression Sum((k*(k + 1))^(m), k=0..N-1) to Faulhaber's formula from Sum((k)^(2*m-1), k=0..N-1) and there holds the identity F(m+i-1,i)=(-1)^i Fe(-m,i), where Fe(-m,i)=A093558(-m,i)/A093559(-m,i) is a Faulhaber coefficient for the sums of even powers of the first N-1 integers (for details see the reference 1, from p. 19).
19, 17, 5, 13, 55, 3, 35, 1, 5, 969969;
21, 57, 17, 21, 13, 33, 63, 7, 5, 63, 4849845; etc.
etc.
Cf. A201453.
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