A093557 Triangle of denominators of coefficients of Faulhaber polynomials in Knuth's version.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 6, 15, 3, 15, 30, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 2, 1, 1, 5, 2, 5, 10, 1, 1, 3, 2, 7, 1, 3, 42, 21, 21, 1, 1, 2, 3, 2, 1, 6, 15, 3, 5, 10, 1, 1, 1, 5, 3, 10, 5, 15, 5, 5, 1, 1, 1, 1, 6, 3, 2, 3, 3, 7, 1, 1, 14, 21, 42, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset

1,5

Comments

The companion triangle with the numerators is A093556, where more information can be found.

Links

Table of n, a(n) for n=1..105.

A. Dzhumadil'daev and D. Yeliussizov, Power sums of binomial coefficients, Journal of Integer Sequences, 16 (2013), Article 13.1.4.

Wolfdieter Lang, First 10 rows.

D. Yeliussizov, Permutation Statistics on Multisets, Ph.D. Dissertation, Computer Science, Kazakh-British Technical University, 2012.

Formula

a(m, k) = denominator(A(m, k)) with recursion: A(m, 0)=1, A(m, k)=-(sum(binomial(m-j, 2*k+1-2*j)*A(m, j), j=0..k-1))/(m-k) if 0<= k <= m-1, else 0. From the 1993 Knuth reference, given in A093556, p. 288, eq.(*) with A^{(m)}_k = A(m, k).

Example

Triangle begins:
  [1];
  [1,1];
  [1,2,1];
  [1,3,3,1];
  ...
Denominators of [1]; [1,0]; [1,-1/2,0]; [1,-4/3,2/3,0]; ... (see W. Lang link in A093556.)

Mathematica

a[m_, k_] := (-1)^(m-k)* Sum[ Binomial[2*m, m-k-j]*Binomial[m-k+j, j]*((m-k-j)/(m-k+j))*BernoulliB[m+k+j], {j, 0, m-k}]; Flatten[ Table[ Denominator[a[m, k]], {m, 1, 14}, {k, 0, m-1}]] (* Jean-François Alcover, Oct 25 2011 *)

Prog

(PARI) T(n, k) = denominator((-1)^(n-k)*sum(j=0, n-k, binomial(2*n, n-k-j)*binomial(n-k+j, j)*(n-k-j)/(n-k+j) * bernfrac(n+k+j))); \\ Michel Marcus, Aug 03 2025

Crossrefs

Cf. A093556 (numerators).

Sequence in context: A129181 A157694 A271187 * A098802 A048804 A158565

Adjacent sequences: A093554 A093555 A093556 * A093558 A093559 A093560

Keyword

nonn,frac,tabl,easy

Author

Wolfdieter Lang, Apr 02 2004

Extensions

More terms from Michel Marcus, Aug 03 2025

Status

approved