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A093559
Triangle of denominators of coefficients of Faulhaber polynomials used for sums of even powers.
4
6, 10, 30, 14, 14, 42, 18, 9, 10, 30, 22, 33, 66, 22, 66, 26, 26, 78, 273, 910, 2730, 30, 30, 15, 9, 90, 2, 6, 34, 51, 51, 51, 102, 51, 170, 510, 38, 19, 95, 95, 190, 57, 3990, 266, 798, 42, 14, 7, 21, 6, 66, 1386, 693, 110, 330, 46, 138, 46, 23, 230, 690, 345, 23, 230, 46
OFFSET
2,1
COMMENTS
The companion triangle with the numerators is A093558. See comment there.
REFERENCES
Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser Verlag, Basel, Boston, Berlin, 1993, ch. 7, pp. 131-159.
LINKS
A. Dzhumadil'daev, D. Yeliussizov, Power sums of binomial coefficients, Journal of Integer Sequences, 16 (2013), Article 13.1.4.
D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comput. 203 (1993), 277-294.
D. Yeliussizov, Permutation Statistics on Multisets, Ph.D. Dissertation, Computer Science, Kazakh-British Technical University, 2012. [N. J. A. Sloane, Jan 03 2013]
FORMULA
a(n, m) = denominator(Fe(m, k), with Fe(m, k):=(m-k)*A(m, k)/(2*m*(2*m-1)) with Faulhaber numbers A(m, k):=A093556(m, k)/A093557(m, k) in Knuth's version. From the bottom of p. 288 of the 1993 Knuth reference.
EXAMPLE
[6]; [10,30]; [14,14,42]; [18,9,10,30]; ...
Denominators of [1/6]; [1/10,-1/30]; [1/14,-1/14,1/42]; [1/18,-1/9,1/10,-1/30]; ... (see W. Lang link in A093558.)
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}]; t[m_, k_] := (m-k)*a[m, k]/(2*m*(2*m-1)); Table[t[m, k] // Denominator, {m, 2, 12}, {k, 0, m-2}] // Flatten (* Jean-François Alcover, Mar 03 2014 *)
CROSSREFS
Sequence in context: A349846 A103767 A025129 * A269697 A271067 A271600
KEYWORD
nonn,frac,tabl,easy
AUTHOR
Wolfdieter Lang, Apr 02 2004
STATUS
approved