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A196840
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Numerators of coefficients of the polynomials in n of the sums of powers of the first n positive integers.
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0
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1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, 0, 1, 1, 1, 0, -1, 0, 5, 1, 1, 1, 0, -1, 0, 1, 1, 1, 0, 1, 0, -7, 0, 7, 1, 1, -1, 0, 2, 0, -7, 0, 2, 1, 1, 0, -3, 0, 1, 0, -7, 0, 3, 1, 1, 5, 0, -1, 0, 1, 0, -1, 0, 5, 1, 1
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OFFSET
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0,19
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COMMENTS
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Duplicate of A162298. - Joerg Arndt, May 27 2019
The sums of the k-th power of each of the first n positive integers, sum(j^k,j=1..n), k>=0, n>=1, abbreviated usually as Sigma n^k, can be written as Sigma n^k = sum(r(k,m)*n^m,m=1..k+1), with the rational number triangle r(n,m)=a(n,m)/A162299(k+1,m). See, e.g., the Graham et al. reference, eq. (6.78), p. 269, where Sigma n^k is S_k(n+1) - delta(k,0), with delta(k,0)=1 if k=0 and 0 else. The formula for r(n,m) given below can be adapted from this reference, and it is found in the given form (for k>0) in the Remmert reference, p. 175.
For sums of powers of integers see the array A103438 with further references and links.
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991 (Seventh printing).Second ed. 1994.
R. Remmert, Funktionentheorie I, Zweite Auflage, Springer-Verlag, 1989. English version: Classical topics in complex function theory, Springer, 1998.
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LINKS
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Table of n, a(n) for n=0..65.
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FORMULA
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a(k,m)= numerator(r(k,m)) with
r(k,m)= 1/(k+1) if m=k+1, 1/2 if m=k, and (B(k+1-m)/(k+1-m))*binomial(k,m) if m = 1,...,k-1, with the Bernoulli numbers B(n)=A027641(n)/A027642(n).
Another version, involving Stirling numbers is
r(k,m) = ((-1)^(k+1-m))*sum(S(k,l)*s(l+1,m)/(l+1),l=(m-1),...,k), k>=0, m=1,...,k+1, with the Stirling numbers of the second kind S, given in A048993, and of the first kind s, given in A048994. For this formula see the W. Lang link under A196837, addendum.
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EXAMPLE
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The triangle a(k,m) starts with
k\m 1 2 3 4 5 6 7 8 9 10 11 ...
0: 1
1: 1 1
2: 1 1 1
3: 0 1 1 1
4: -1 0 1 1 1
5: 0 -1 0 5 1 1
6: 1 0 -1 0 1 1 1
7: 0 1 0 -7 0 7 1 1
8: -1 0 2 0 -7 0 2 1 1
9: 0 -3 0 1 0 -7 0 3 1 1
10: 5 0 -1 0 1 0 -1 0 5 1 1
...
The rational number triangle a(k,m)/A162299(k+1,m) starts with
k\m 1 2 3 4 5 6 7 8 9 ...
0: 1
1: 1/2 1/2
2: 1/6 1/2 1/3
3: 0 1/4 1/2 1/4
4: -1/30 0 1/3 1/2 1/5
5: 0 -1/12 0 5/12 1/2 1/6
6: 1/42 0 -1/6 0 1/2 1/2 1/7
7: 0 1/12 0 -7/24 0 7/12 1/2 1/8
8: -1/30 0 2/9 0 -7/15 0 2/3 1/2 1/9
...
Sigma n^4 = sum(j^4,j=1..n) =
-(1/30)*n + (1/3)*n^3 + (1/2)*n^4 + (1/5)*n^5.
For n>=1 this is the sequence A000538(n).
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MATHEMATICA
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row[k_] := Numerator[ Rest[ CoefficientList[ HarmonicNumber[n, -k], n]]]; Flatten[ Table[ row[k], {k, 0, 10}]] (* Jean-François Alcover, Dec 07 2011 *)
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CROSSREFS
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A103438, A196837.
Sequence in context: A086464 A227577 A281446 * A162298 A196755 A199510
Adjacent sequences: A196837 A196838 A196839 * A196841 A196842 A196843
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KEYWORD
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sign,easy,tabl
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AUTHOR
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Wolfdieter Lang, Oct 23 2011
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STATUS
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approved
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