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A093558
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Triangle of numerators of coefficients of Faulhaber polynomials used for sums of even powers.
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3
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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; internal format)
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OFFSET
| 2,12
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COMMENTS
| The companion triangle with the denominators is A093559.
Sum(k^(2*(m-1)),k=1..n) = (2*n+1)*sum( Fe(m,k)*(n*(n+1))^(m-1-j),j=0..m-1), 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.
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REFERENCES
| D. E. Knuth, Johann Faulhaber and sums of powers, Maths. of Computation 61, 203 (1993) 277-294.
Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhaeuser Verlag, Basel, Boston, Berlin, 1993, ch.7, p. 131-159.
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LINKS
| W. Lang, First 10 rows and triangle with rational entries.
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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.
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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)
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CROSSREFS
| Sequence in context: A160739 A092679 A090592 * A170866 A125636 A156323
Adjacent sequences: A093555 A093556 A093557 * A093559 A093560 A093561
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KEYWORD
| sign,frac,tabl,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Apr 02 2004
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