1,8

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^(2*m-1),j=1..n)= sum(A(m,k)*u^(m-k),k=0..m-1)/(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^(2*(m-1)),j=1..n)= (2*n+1)*sum((m-j)*A(m,j)*(n*(n+1))^(m-1-j),j=0..m-1)/(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.

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, Maths. of Computation 61, 203 (1993) 277-294.

Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhaeuser Verlag, Basel, Boston, Berlin, 1993, ch.7, p. 131-159.

W. Lang, First 10 rows and Faulhaber triangle with rational entries and examples.

a(m, k)= numerator(A(m, k)) with recursion: A(m, 0)=1, A(m, k)=-(sum(binomial(m-j, 2*k+1-2*j)*A(m, j), j=0..k-1))/(m-k) if 0<= k <= m-1, else 0. From the Knuth 1993 reference, p. 288, eq.(*) with A^{(m)}_k = A(m, k).

A(m, k)=((-1)^(m-k))*sum(binomial(2*m, m-k-j)*binomial(m-k+j, j)*((m-k-j)/(m-k+j))*Bernoulli(m+k+j), j=0..m-k). 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.

[1]; [1,0]; [1,-1,0]; [1,-4,2,0]; ...

Numerators of Knuth's Faulhaber triangle A(m,k):

rows [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:

rows [1/2], [ -1/6, 1/3], [1/6, -1/3, 1/4], [ -3/10, 3/5, -1/2, 1/5],... for m>=2, l>=2.

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: A137986 A093486 A115143 this_sequence A021242 A088393 A121225

Adjacent sequences: A093553 A093554 A093555 this_sequence A093557 A093558 A093559

sign,frac,tabl,easy

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Apr 02 2004