A093558 - OEIS

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

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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

Table of n, a(n) for n=2..56.

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]

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

Adjacent sequences: A093555 A093556 A093557 * A093559 A093560 A093561

KEYWORD

sign,frac,tabl,easy

AUTHOR

Wolfdieter Lang, Apr 02 2004

STATUS

approved