A093556 Triangle of numerators of coefficients of Faulhaber polynomials in Knuth's version.

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

Author

Wolfdieter Lang, Apr 02 2004

Keywords

• sign
• frac
• tabl
• easy

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.

Examples

			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],
  ...
		

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.
• Wolfdieter Lang, First 10 rows and Faulhaber triangle with rational entries and examples.
• D. Yeliussizov, Permutation Statistics on Multisets, Ph.D. Dissertation, Computer Science, Kazakh-British Technical University, 2012. - _N. J. A. Sloane_, Jan 03 2013

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.

Cf. A027641, A027642, A093558, A093559, A103438.

Programs

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 *)

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

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.

Extensions

More terms from Michel Marcus, Aug 03 2025