login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A093557
Triangle of denominators of coefficients of Faulhaber polynomials in Knuth's version.
8
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
OFFSET
1,5
COMMENTS
The companion triangle with the numerators is A093556, where more information can be found.
LINKS
A. Dzhumadil'daev, D. Yeliussizov, Power sums of binomial coefficients, Journal of Integer Sequences, 16 (2013), Article 13.1.4.
W. 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 *)
CROSSREFS
Sequence in context: A129181 A157694 A271187 * A098802 A048804 A158565
KEYWORD
nonn,frac,tabl,easy
AUTHOR
Wolfdieter Lang, Apr 02 2004
STATUS
approved