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.)
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 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Apr 02 2004
STATUS
approved