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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 Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser Verlag, Basel, Boston, Berlin, 1993, ch. 7, pp. 131-159. LINKS A. Dzhumadil'daev, 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 [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 January 29 16:58 EST 2020. Contains 331347 sequences. (Running on oeis4.)