A093558 Triangle of numerators of coefficients of Faulhaber polynomials used for sums of even powers.

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

Author

Wolfdieter Lang, Apr 02 2004

Keywords

• sign
• frac
• tabl
• easy

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.

Examples

			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)
		

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.
• Wolfdieter Lang, First 10 rows and triangle with rational entries.
• D. Yeliussizov, Permutation Statistics on Multisets, Ph.D. Dissertation, Computer Science, Kazakh-British Technical University, 2012. [_N. J. A. Sloane_, Jan 03 2013]

Crossrefs

Cf. A093556/A093557, A093559 (denominators).

Programs

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 *)

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.