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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Damir Yeliussizov

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Damir Yeliussizov has authored 2 sequences.

A201453 Triangle of numerators of dual coefficients of Faulhaber.

Original entry on oeis.org

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

Views

Author

Damir Yeliussizov, Jan 09 2013

Keywords

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

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)

A201454 Triangle of denominators of dual coefficients of Faulhaber.

Original entry on oeis.org

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

Views

Author

Damir Yeliussizov, Jan 09 2013

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

Cf. A201453.

Programs

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

Formula

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).

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