A093559 Triangle of denominators of coefficients of Faulhaber polynomials used for sums of even powers.

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

Table of n, a(n) for n=2..66.

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

Adjacent sequences: A093556 A093557 A093558 * A093560 A093561 A093562

Keyword

nonn,frac,tabl,easy

Author

Wolfdieter Lang, Apr 02 2004

Status

approved