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A093556
Triangle of numerators of coefficients of Faulhaber polynomials in Knuth's version.
10
1, 1, 0, 1, -1, 0, 1, -4, 2, 0, 1, -5, 3, -3, 0, 1, -4, 17, -10, 5, 0, 1, -35, 287, -118, 691, -691, 0, 1, -8, 112, -352, 718, -280, 140, 0, 1, -21, 66, -293, 4557, -3711, 10851, -10851, 0, 1, -40, 217, -4516, 2829, -26332, 750167, -438670, 219335, 0, 1, -33, 506, -2585, 7579, -198793, 1540967, -627073, 1222277, -1222277, 0
OFFSET
1,8
COMMENTS
The companion triangle with the denominators is A093557.
In the 1986 Edwards reference, eq. 7, p. 453, the lower triangular matrix F^{-1} is obtained from F^{-1}(m,l) = A(m,m-l)/m with m >= 2, l >= 2. See the W. Lang link for this triangle.
Sum_{j=1..n} j^(2*m-1) = Sum_{k=0..m-1} A(m,k)*u^(m-k)/(2*m), with u:=n*(n+1), A(m,k):= A093556(m,k)/ A093557(m,k) and m=1,2,... (Faulhaber's m-th row polynomial in falling powers of u:=n*(n+1), divided by 2*m, gives the sum of the (2*m-1)-th power of the first n integers > 0. See the W. Lang link for the Faulhaber triangle.)
Sum_{j=1..n} j^(2*(m-1)) = (2*n+1)*Sum_{j=0..m-1} (m-j)*A(m,j)*(n*(n+1))^(m-1-j)/(2*m*(2*m-1)), with u:=n*(n+1) and m >= 2. Sum of the even powers of the first n integers > 0. From the bottom of p. 288 of the 1993 Knuth reference with A^{(m)}_k = A(m,k). See also A093558 with A093559.
REFERENCES
Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser Verlag, Basel, Boston, Berlin, 1993, ch. 7, pp. 131-159.
LINKS
A. W. F. Edwards, A quick route to sums of powers, Amer. Math. Monthly 93 (1986) 451-455.
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(m, k) = numerator(A(m, k)) with recursion: A(m, 0)=1, A(m, k) = -(Sum_{j=0..k-1} binomial(m-j, 2*k+1-2*j)*A(m, j))/(m-k) if 0 <= k <= m-1, otherwise 0. From the Knuth 1993 reference, p. 288, eq.(*) with A^{(m)}_k = A(m, k).
A(m, k) = ((-1)^(m-k))*Sum_{j=0..m-k} binomial(2*m, m-k-j)*binomial(m-k+j, j)*((m-k-j)/(m-k+j))*Bernoulli(m+k+j). From the Knuth 1993 reference, p. 289, last eq. with A^{(m)}_k = A(m, k). Attributed to I. M. Gessel and X. G. Viennot (see A065551 for the 1989 reference). For Bernoulli numbers see A027641 with A027642.
EXAMPLE
Triangle begins:
[1];
[1,0];
[1,-1,0];
[1,-4,2,0];
...
Numerators of Knuth's Faulhaber triangle A(m,k):
[1],
[1, 0],
[1, -1/2, 0],
[1, -4/3, 2/3, 0],
...
A(m,m-1)=1 if m=1, else 0.
Edwards' Faulhaber triangle F^{-1}(m,l) = A(m,m-l)/m, for m>=2, l>=2:
[1/2],
[-1/6, 1/3],
[1/6, -1/3, 1/4],
[-3/10, 3/5, -1/2, 1/5],
...
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}]; Flatten[ Table[ Numerator[a[m, k]], {m, 1, 11}, {k, 0, m-1}]] (* Jean-François Alcover, Oct 25 2011 *)
PROG
(PARI) T(n, k) = numerator((-1)^(n-k)*sum(j=0, n-k, binomial(2*n, n-k-j)*binomial(n-k+j, j)*(n-k-j)/(n-k+j) * bernfrac(n+k+j))); \\ Michel Marcus, Aug 03 2025
CROSSREFS
Cf. A093557 (denominators).
Cf. A065551 and A065553 for Ira M. Gessel's and X. G. Viennot's version of Faulhaber triangle which is Edwards' Faulhaber triangle augmented with a first row and first column.
Sequence in context: A377036 A354681 A115143 * A021242 A088393 A121225
KEYWORD
sign,frac,tabl,easy
AUTHOR
Wolfdieter Lang, Apr 02 2004
EXTENSIONS
More terms from Michel Marcus, Aug 03 2025
STATUS
approved