cp's OEIS Frontend

A196840 Duplicate of A162298.

%I A196840 #22 Mar 17 2025 14:56:47
%S A196840 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,
%T A196840 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,
%U A196840 5,1,1
%N A196840 Duplicate of A162298.
%C A196840 Duplicate of A162298. - _Joerg Arndt_, May 27 2019
%C A196840 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.
%C A196840 For sums of powers of integers see the array A103438 with further references and links.
%D A196840 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991 (Seventh printing).Second ed. 1994.
%D A196840 R. Remmert, Funktionentheorie I, Zweite Auflage, Springer-Verlag, 1989. English version: Classical topics in complex function theory, Springer, 1998.
%F A196840 a(k,m)= numerator(r(k,m)) with
%F A196840 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).
%F A196840 Another version, involving Stirling numbers is
%F A196840 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.
%e A196840 The triangle a(k,m) starts with
%e A196840 k\m  1   2  3  4  5  6  7  8  9 10 11 ...
%e A196840 0:   1
%e A196840 1:   1   1
%e A196840 2:   1   1  1
%e A196840 3:   0   1  1  1
%e A196840 4:  -1   0  1  1  1
%e A196840 5:   0  -1  0  5  1  1
%e A196840 6:   1   0 -1  0  1  1  1
%e A196840 7:   0   1  0 -7  0  7  1  1
%e A196840 8:  -1   0  2  0 -7  0  2  1  1
%e A196840 9:   0  -3  0  1  0 -7  0  3  1  1
%e A196840 10:  5   0 -1  0  1  0 -1  0  5  1  1
%e A196840 ...
%e A196840 The rational number triangle a(k,m)/A162299(k+1,m) starts with
%e A196840 k\m  1     2     3     4     5      6    7    8    9 ...
%e A196840 0:   1
%e A196840 1:  1/2   1/2
%e A196840 2:  1/6   1/2   1/3
%e A196840 3:   0    1/4   1/2   1/4
%e A196840 4: -1/30   0    1/3   1/2   1/5
%e A196840 5:   0   -1/12   0   5/12   1/2    1/6
%e A196840 6:  1/42   0   -1/6    0    1/2    1/2  1/7
%e A196840 7:   0    1/12   0   -7/24   0    7/12  1/2  1/8
%e A196840 8: -1/30   0    2/9    0   -7/15    0   2/3  1/2  1/9
%e A196840 ...
%e A196840 Sigma n^4 = sum(j^4,j=1..n) =
%e A196840 -(1/30)*n + (1/3)*n^3 +  (1/2)*n^4 + (1/5)*n^5.
%e A196840 For n>=1 this is the sequence A000538(n).
%t A196840 row[k_] := Numerator[ Rest[ CoefficientList[ HarmonicNumber[n, -k], n]]]; Flatten[ Table[ row[k], {k, 0, 10}]] (* _Jean-François Alcover_, Dec 07 2011 *)
%Y A196840 A103438, A196837.
%K A196840 dead
%O A196840 0,19
%A A196840 _Wolfdieter Lang_, Oct 23 2011