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A201454 Triangle of denominators of dual coefficients of Faulhaber.

%I A201454 #30 Sep 08 2022 08:46:01
%S A201454 1,3,3,5,3,15,7,5,3,105,9,21,15,9,105,11,9,21,3,9,231,13,11,3,7,5,3,
%T A201454 15015,15,39,165,9,15,5,45,2145,17,5,13,55,9,15,15,45,36465,19,17,5,
%U A201454 13,55,3,35,1,5,969969,21,57,17,21,13,33,63,7,5,63,4849845
%N A201454 Triangle of denominators of dual coefficients of Faulhaber.
%C A201454 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 A201454 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 link, from p. 19).
%H A201454 Askar Dzhumadil'daev, 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
%F A201454 a(m,k) = denominator(F(m,k)) with F(m,k) = (1/(2*m-2*k+1)) * sum(i=0..2*k, binomial(m,2*k-i) * binomial(2*m-2*k+i,i) * Bernoulli(i) ).
%F A201454 A recursion is given by 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).
%e A201454 Triangle begins:
%e A201454 1;
%e A201454 3,  3;
%e A201454 5,  3,  15;
%e A201454 7,  5,  3,   10;
%e A201454 9,  21, 15,  9,  105;
%e A201454 11, 9,  21,  3,  9,   231;
%e A201454 13, 11, 3,   7,  5,   3,   15015;
%e A201454 15, 39, 165, 9,  15,  5,   45,    2145;
%e A201454 17, 5,  13,  55, 9,   15,  15,    45,   36465;
%e A201454 19, 17, 5,   13, 55,  3,   35,    1,    5,    969969;
%e A201454 21, 57, 17,  21, 13,  33,  63,    7,    5,    63,    4849845;
%e A201454 etc.
%t A201454 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 A201454 a[m_, k_] := f[m, k] // Denominator;
%t A201454 Table[a[m, k], {m, 0, 10}, {k, 0, m}] // Flatten (* _Jean-François Alcover_, Jan 18 2013 *)
%o A201454 (Magma) [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
%Y A201454 Cf. A201453.
%K A201454 nonn,frac,tabl,easy
%O A201454 0,2
%A A201454 _Damir Yeliussizov_, Jan 09 2013