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A093557 Triangle of denominators of coefficients of Faulhaber polynomials in Knuth's version. 7
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 6, 15, 3, 15, 30, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 2, 1, 1, 5, 2, 5, 10, 1, 1, 3, 2, 7, 1, 3, 42, 21, 21, 1, 1, 2, 3, 2, 1, 6, 15, 3, 5, 10, 1, 1, 1, 5, 3, 10, 5, 15, 5, 5, 1, 1, 1, 1, 6, 3, 2, 3, 3, 7, 1, 1, 14, 21, 42, 1, 1, 1, 2, 1, 1, 1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

The companion triangle with the numerators is A093556, where more information can be found.

REFERENCES

Askar Dzhumadildaev and Damir Yeliussizov, "Power Sums of Binomial Coefficients", Journal of Integer Sequences, Vol. 16 (2013), #13.1.1.

D. Yeliussizov, Permutation Statistics on Multisets, Ph.D. Dissertation, Computer Science, Kazakh-British Technical University, 2012; http://www.kazntu.kz/sites/default/files/20121221ND_Eleusizov.pdf.

LINKS

Table of n, a(n) for n=1..99.

W. Lang, First 10 rows.

FORMULA

a(m, k)= denominator(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 1993 Knuth reference, given in A093556, p. 288, eq.(*) with A^{(m)}_k = A(m, k).

EXAMPLE

[1]; [1,1]; [1,2,1]; [1,3,3,1]; ...

Denominators of [1]; [1,0]; [1,-1/2,0]; [1,-4/3,2/3,0]; ... (see W. Lang link in A093556.)

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[ Denominator[a[m, k]], {m, 1, 14}, {k, 0, m-1}]] (* Jean-Fran├žois Alcover, Oct 25 2011 *)

CROSSREFS

Sequence in context: A129181 A157694 A271187 * A098802 A048804 A158565

Adjacent sequences:  A093554 A093555 A093556 * A093558 A093559 A093560

KEYWORD

nonn,frac,tabl,easy

AUTHOR

Wolfdieter Lang, Apr 02 2004

STATUS

approved

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Last modified February 20 21:53 EST 2018. Contains 299387 sequences. (Running on oeis4.)