A201453 -id:A201453 - cp's OEIS frontend

A201454 Triangle of denominators of dual coefficients of Faulhaber.

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, 15015, 15, 39, 165, 9, 15, 5, 45, 2145, 17, 5, 13, 55, 9, 15, 15, 45, 36465, 19, 17, 5, 13, 55, 3, 35, 1, 5, 969969, 21, 57, 17, 21, 13, 33, 63, 7, 5, 63, 4849845

Offset: 0

Damir Yeliussizov, Jan 09 2013

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,...

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).

			Triangle begins:
1;
3,  3;
5,  3,  15;
7,  5,  3,   10;
9,  21, 15,  9,  105;
11, 9,  21,  3,  9,   231;
13, 11, 3,   7,  5,   3,   15015;
15, 39, 165, 9,  15,  5,   45,    2145;
17, 5,  13,  55, 9,   15,  15,    45,   36465;
19, 17, 5,   13, 55,  3,   35,    1,    5,    969969;
21, 57, 17,  21, 13,  33,  63,    7,    5,    63,    4849845;
etc.

Askar Dzhumadil'daev, Damir Yeliussizov, Power sums of binomial coefficients, Journal of Integer Sequences, 16 (2013), Article 13.1.6

Cf. A201453.

[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

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] // Denominator;
Table[a[m, k], {m, 0, 10}, {k, 0, m}] // Flatten (* Jean-François Alcover, Jan 18 2013 *)

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) ).

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).

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

A201454 Triangle of denominators of dual coefficients of Faulhaber.

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