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A093557
Triangle of denominators of coefficients of Faulhaber polynomials in Knuth's version.
10
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
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
KEYWORD
nonn,frac,tabl,easy
AUTHOR
Wolfdieter Lang, Apr 02 2004
EXTENSIONS
More terms from Michel Marcus, Aug 03 2025
STATUS
approved