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A093558 Triangle of numerators of coefficients of Faulhaber polynomials used for sums of even powers. 5
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, 1, -55, 209, -902, 60511 (list; table; graph; refs; listen; history; text; internal format)
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

Askar Dzhumadildaev and Damir Yeliussizov, "Power Sums of Binomial Coefficients", Journal of Integer Sequences, Vol. 16 (2013), #13.1.1.

D. E. Knuth, Johann Faulhaber and sums of powers, Maths. of Computation 61, 203 (1993) 277-294.

Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser Verlag, Basel, Boston, Berlin, 1993, ch. 7, pp. 131-159.

D. Yeliussizov, Permutation Statistics on Multisets, Ph.D. Dissertation, Computer Science, Kazakh-British Technical University, 2012; http://www.kazntu.kz/sites/default/files/20121221ND_Eleusizov.pdf.-From N. J. A. Sloane, Jan 03 2013

LINKS

Table of n, a(n) for n=2..61.

W. Lang, First 10 rows and triangle with rational entries.

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

[1]; [1,-1]; [1,-1,1]; [1,-1,1,-1]; ...

Numerators 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)

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

Sequence in context: A092679 A277534 A090592 * A170866 A125636 A156323

Adjacent sequences:  A093555 A093556 A093557 * A093559 A093560 A093561

KEYWORD

sign,frac,tabl,easy

AUTHOR

Wolfdieter Lang, Apr 02 2004

STATUS

approved

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Last modified November 13 19:25 EST 2018. Contains 317149 sequences. (Running on oeis4.)