VOOZH about

URL: https://oeis.org/A093558

⇱ A093558 - OEIS

login
A093558
Triangle of numerators of coefficients of Faulhaber polynomials used for sums of even powers.
7
1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -5, 17, -5, 5, 1, -5, 41, -236, 691, -691, 1, -7, 14, -22, 359, -7, 7, 1, -14, 77, -293, 1519, -1237, 3617, -3617, 1, -6, 217, -1129, 8487, -6583, 750167, -43867, 43867, 1, -5, 23, -470, 689, -28399, 1540967, -1254146, 174611, -174611
OFFSET
2,12
COMMENTS
The companion triangle with the denominators is A093559.
Sum_{k=1..n} k^(2*(m-1)) = (2*n+1)*Sum_{j=0..m-1} Fe(m,k)*(n*(n+1))^(m-1-j), m >= 2. Sums of even powers of the first n integers >0 as polynomials in u := n*(n+1) (falling powers of u). See bottom of p. 288 of the 1993 Knuth reference.
REFERENCES
Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser Verlag, Basel, Boston, Berlin, 1993, ch. 7, pp. 131-159.
LINKS
A. Dzhumadil'daev and 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) = numerator(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
Triangle begins:
[1];
[1,-1];
[1,-1,1];
[1,-1,1,-1];
[1,-5,17,-5,5]
...
Numerators of:
[1/6];
[1/10,-1/30];
[1/14,-1/14,1/42];
[1/18,-1/9,1/10,-1/30];
[1/22,-5/33,17/66,-5/22,5/66];
... (see Lang link)
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] // Numerator, {m, 2, 12}, {k, 0, m-2}] // Flatten (* Jean-François Alcover, Mar 03 2014 *)
CROSSREFS
Cf. A093556/A093557, A093559 (denominators).
Sequence in context: A090592 A377618 A340706 * A170866 A393017 A337031
KEYWORD
sign,frac,tabl,easy
AUTHOR
Wolfdieter Lang, Apr 02 2004
STATUS
approved