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A201453 Triangle of numerators of dual coefficients of Faulhaber.

%I A201453 #31 Aug 09 2025 11:23:38
%S A201453 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 A201453 -8,6112,1,-28,602,-100,313,-112,512,-3712,1,-4,70,-1268,593,-1816,
%U A201453 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 A201453 Triangle of numerators of dual coefficients of Faulhaber.
%C A201453 Sum_{k=0..N-1} (k*(k + 1))^m = Sum_{i=0..m} F(m,i)*N^(2*m-2*i+1), m=0,1,2,...
%C A201453 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 A201453 A. Dzhumadil'daev and 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 A201453 a(m,k) = numerator(F(m,k)) with:
%F A201453 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 A201453 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 A201453 Triangle begins:
%e A201453   1;
%e A201453   1, -1;
%e A201453   1, -1,  2;
%e A201453   1, -2,  1,   -8;
%e A201453   1, -10, 11,  -4,    8;
%e A201453   1, -5,  29,  -5,    8,    -32;
%e A201453   1, -7,  7,   -33,   26,   -8,    6112;
%e A201453   1, -28, 602, -100,  313,  -112,  512,   -3712;
%e A201453   1, -4,  70,  -1268, 593,  -1816, 1936,  -2944, 362624;
%e A201453   1, -15, 38,  -566,  9681, -1481, 31568, -960,  2432,   -71706112;
%e A201453   ...
%t A201453 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 A201453 a[m_, k_] := f[m, k] // Numerator;
%t A201453 Table[a[m, k], {m, 0, 10}, {k, 0, m}] // Flatten
%o A201453 (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 A201453 Cf. A093558, A093559, A201454 (denominators).
%K A201453 sign,frac,tabl,easy
%O A201453 0,6
%A A201453 _Damir Yeliussizov_, Jan 09 2013