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A162298 Faulhaber's triangle: triangle T(k,y) read by rows, giving numerator of the coefficient [m^y] of the polynomial Sum_{x=1..m} x^(k-1).

%I A162298 #61 Mar 17 2025 14:54:56
%S A162298 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 A162298 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 A162298 5,1,1,0,5,0,-11,0,11,0,-11,0,11,1,1,-691,0,5,0,-33,0,22,0,-11,0,1,1,1,0,-691,0,65,0,-143,0,143,0,-143,0,13,1,1
%N A162298 Faulhaber's triangle: triangle T(k,y) read by rows, giving numerator of the coefficient [m^y] of the polynomial Sum_{x=1..m} x^(k-1).
%C A162298 There are many versions of Faulhaber's triangle: search the OEIS for his name. For example, A220962/A220963 is essentially the same as this triangle, except for an initial column of 0's. - _N. J. A. Sloane_, Jan 28 2017
%C A162298 Named after the German mathematician Johann Faulhaber (1580-1653). - _Amiram Eldar_, Jun 13 2021
%C A162298 From _Wolfdieter Lang_, Oct 23 2011 (Start):
%C A162298 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 A162298 For sums of powers of integers see the array A103438 with further references and links.
%C A162298 (End)
%H A162298 Alois P. Heinz, <a href="/A162298/b162298.txt">Rows n = 0..140, flattened</a>
%H A162298 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991 (Seventh printing).Second ed. 1994.
%H A162298 R. Remmert, Funktionentheorie I, Zweite Auflage, Springer-Verlag, 1989. English version: Classical topics in complex function theory, Springer, 1998.
%H A162298 Mohammad Torabi-Dashti, <a href="http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/faulhaber-s-triangle">Faulhaber's Triangle</a>, College Math. J., Vol. 42, No. 2 (2011), pp. 96-97.
%H A162298 Mohammad Torabi-Dashti, <a href="/A162298/a162298.pdf">Faulhaber’s Triangle</a>. [Annotated scanned copy of preprint]
%H A162298 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/PowerSum.html">Power Sum</a>.
%F A162298 Faulhaber's triangle of fractions H(n,k) (n >= 0, 1 <= k <= n+1) is defined by: H(0,1)=1; for 2<=k<=n+1, H(n,k) = (n/k)*H(n-1,k-1) with H(n,1) =  1 - Sum_{i=2..n+1}H(n,i). - _N. J. A. Sloane_, Jan 28 2017
%F A162298 Sum_{x=1..m} x^(k-1) = (Bernoulli(k,m+1)-Bernoulli(k))/k.
%F A162298 T(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). Alternatively 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 S given in A048993, and s given in A048994. - _Wolfdieter Lang_, Oct 23 2011
%e A162298 The first few polynomials:
%e A162298     m;
%e A162298    m/2  + m^2/2;
%e A162298    m/6  + m^2/2 + m^3/3;
%e A162298     0   + m^2/4 + m^3/2 + m^4/4;
%e A162298   -m/30 +   0   + m^3/3 + m^4/2 + m^5/5;
%e A162298   ...
%e A162298 Initial rows of Faulhaber's triangle of fractions H(n, k) (n >= 0, 1 <= k <= n+1):
%e A162298     1;
%e A162298    1/2,  1/2;
%e A162298    1/6,  1/2,  1/3;
%e A162298     0,   1/4,  1/2,  1/4;
%e A162298   -1/30,  0,   1/3,  1/2,  1/5;
%e A162298     0,  -1/12,  0,   5/12, 1/2,  1/6;
%e A162298    1/42,  0,  -1/6,   0,   1/2,  1/2,  1/7;
%e A162298     0,   1/12,  0,  -7/24,  0,   7/12, 1/2,  1/8;
%e A162298   -1/30,  0,   2/9,   0,  -7/15,  0,   2/3,  1/2,  1/9;
%e A162298   ...
%p A162298 A162298 := proc(k, y) local gf, x; gf := sum(x^(k-1), x=1..m) ; coeftayl(gf, m=0, y) ; numer(%) ; end proc: # _R. J. Mathar_, Mar 26 2013
%p A162298 # To produce Faulhaber's triangle of fractions H(n,k) (n >= 0, 1 <= k <= n+1):
%p A162298 H:=proc(n,k) option remember; local i;
%p A162298 if n<0 or k>n+1 then 0;
%p A162298 elif n=0 then 1;
%p A162298 elif k>1 then (n/k)*H(n-1,k-1);
%p A162298 else 1 - add(H(n,i),i=2..n+1); fi; end;
%p A162298 for n from 0 to 10 do lprint([seq(H(n,k),k=1..n+1)]); od:
%p A162298 for n from 0 to 12 do lprint([seq(numer(H(n,k)),k=1..n+1)]); od: # A162298
%p A162298 for n from 0 to 12 do lprint([seq(denom(H(n,k)),k=1..n+1)]); od: # A162299 # _N. J. A. Sloane_, Jan 28 2017
%t A162298 H[n_, k_] := H[n, k] = Which[n < 0 || k > n+1, 0, n == 0, 1, k > 1, (n/k)* H[n-1, k-1], True, 1 - Sum[H[n, i], {i, 2, n+1}]];
%t A162298 Table[H[n, k] // Numerator, {n, 0, 13}, {k, 1, n+1}] // Flatten (* _Jean-François Alcover_, Aug 04 2022 *)
%Y A162298 Cf. A000367, A162299 (denominators), A103438, A196837.
%Y A162298 See also A220962/A220963.
%K A162298 tabl,frac,sign
%O A162298 0,19
%A A162298 _Juri-Stepan Gerasimov_, Jun 30 2009 and Jul 02 2009
%E A162298 Offset set to 0 by _Alois P. Heinz_, Feb 19 2021