A201454 -id:A201454 - cp's OEIS frontend

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

Damir Yeliussizov, Jan 09 2013

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).

			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;
  ...

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

Cf. A093558, A093559, A201454 (denominators).

[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

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

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)

A230069 Numerators of inverse of triangle A082985(n).

Original entry on oeis.org

1, -1, 1, 2, -1, 1, -8, 1, -2, 1, 8, -4, 11, -10, 1, -32, 8, -5, 29, -5, 1, 6112, -8, 26, -33, 7, -7, 1, -3712, 512, -112, 313, -100, 602, -28, 1, 362624, -2944, 1936, -1816, 593, -1268, 70, -4, 1, -71706112, 2432, -960, 31568, -1481, 9681, -566, 38, -15, 1

Offset: 0

Paul Curtz, Oct 08 2013

First column of the example: A212196(n)/A181131(n), main diagonal of A164555(n)/A027642(n). See A190339(n). Hence a link between Chebyshev and Bernoulli numbers.

Mirror image of A201453.

			Numerators of
1,
-1/3,    1/3,
2/15,   -1/3,   1/5,
-8/105,  1/3,  -2/5,    1/7,
8/105,  -4/9, 11/15, -10/21,  1/9,
-32/231, 8/9,  -5/3,  29/21, -5/9, 1/11

Cf. A201453(n)/A201454(n), A098435.

rows = 10; u[n_, m_] /; m > n = 0; u[n_, m_] := Binomial[2*n - m, m]*(2*n + 1)/(2*n - 2*m + 1); t = Table[u[n, m], {n, 0, rows - 1}, {m, 0, rows - 1}] // Inverse; Table[t[[n, k]] // Numerator, {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 08 2013 *)

T(k,m) = numerator of F(k,m) = (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)). - Ralf Stephan, Oct 10 2013

More terms from Jean-François Alcover, Oct 08 2013

cp's OEIS Frontend

A201453 Triangle of numerators of dual coefficients of Faulhaber.

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