%I #23 Sep 08 2022 08:46:01

%S 1,1,-1,1,-1,2,1,-2,1,-8,1,-10,11,-4,8,1,-5,29,-5,8,-32,1,-7,7,-33,26,

%T -8,6112,1,-28,602,-100,313,-112,512,-3712,1,-4,70,-1268,593,-1816,

%U 1936,-2944,362624,1,-15,38,-566,9681,-1481,31568,-960,2432,-71706112,1,-55,176,-1606,5401,-54499,290362,-58864,44736,-285568,3341113856

%N Triangle of numerators of dual coefficients of Faulhaber

%C Sum((k*(k + 1))^(m), k=0..N-1)=Sum(F(m,i)*N^(2*m-2*i+1),i=0..m), m=0,1,2,...

%C 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).

%H A. Dzhumadil'daev, D. 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

%F a(m,k)=numerator(F(m,k)) with:

%F 1) recursion, F(x,0) = 1/(2*x+1) and 2*(m-k)*(2*m-2*k+1)*F(m,k)=2*m*(2*m-1)*F(m-1,k)+m*(m-1)*F(m-2,k-1);

%F 2) explicit formula F(m,k) = (1/(2*m-2*k+1))sum(binomial(m,2*k-i)*binomial(2*m-2*k+i,i) Bernoulli(i), i=0..2*k)

%e Triangle begins:

%e 1;

%e 1, -1;

%e 1, -1, 2;

%e 1, -2, 1, -8;

%e 1, -10, 11, -4, 8;

%e 1, -5, 29, -5, 8, -32;

%e 1, -7, 7, -33, 26, -8, 6112;

%e 1, -28, 602, -100, 313, -112, 512, -3712;

%e 1, -4, 70, -1268, 593, -1816, 1936, -2944, 362624;

%e 1, -15, 38, -566, 9681, -1481, 31568, -960, 2432, -71706112; etc.

%t f[m_, k_] := (1/(2*m - 2*k + 1))* Sum[Binomial[m, 2*k - i]*Binomial[2*m - 2*k + i, i]*BernoulliB[i], {i, 0, 2 k}];

%t a[m_, k_] := f[m, k] // Numerator;

%t Table[a[m, k], {m, 0, 10}, {k, 0, m}] // Flatten

%o (Magma) [Numerator((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

%Y Cf. A093558, A093559, A201454 (denominators).

%K sign,frac,tabl,easy

%O 0,6

%A _Damir Yeliussizov_, Jan 09 2013