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

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

Table of n, a(n) for n=1..63.

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.

W. 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

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 *)

Crossrefs

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. A103438.

Sequence in context: A259618 A354681 A115143 * A021242 A088393 A121225

Adjacent sequences: A093553 A093554 A093555 * A093557 A093558 A093559

Keyword

sign,frac,tabl,easy

Author

Wolfdieter Lang, Apr 02 2004

Status

approved