A201453 Triangle of numerators of dual coefficients of Faulhaber.

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, -8, 6112, 1, -28, 602, -100, 313, -112, 512, -3712, 1, -4, 70, -1268, 593, -1816, 1936, -2944, 362624, 1, -15, 38, -566, 9681, -1481, 31568, -960, 2432, -71706112, 1, -55, 176, -1606, 5401, -54499, 290362, -58864, 44736, -285568, 3341113856

Offset: 0

Author

Damir Yeliussizov, Jan 09 2013

Keywords

• sign
• frac
• tabl
• easy

Comments

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,...

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).

Examples

			Triangle begins:
  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,   -8,    6112;
  1, -28, 602, -100,  313,  -112,  512,   -3712;
  1, -4,  70,  -1268, 593,  -1816, 1936,  -2944, 362624;
  1, -15, 38,  -566,  9681, -1481, 31568, -960,  2432,   -71706112;
  ...
		

Links

• A. Dzhumadil'daev and D. Yeliussizov, Power sums of binomial coefficients, Journal of Integer Sequences, 16 (2013), Article 13.1.6

Crossrefs

Cf. A093558, A093559, A201454 (denominators).

Programs

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

Mathematica

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}];
a[m_, k_] := f[m, k] // Numerator;
Table[a[m, k], {m, 0, 10}, {k, 0, m}] // Flatten

Formula

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

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);

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)