A080779 Triangle read by rows: n-th row gives expansion of the series for HarmonicNumber(n, -r).

1, 1, 1, 1, 3, 2, 0, 6, 12, 6, -4, 0, 40, 60, 24, 0, -60, 0, 300, 360, 120, 120, 0, -840, 0, 2520, 2520, 720, 0, 3360, 0, -11760, 0, 23520, 20160, 5040, -12096, 0, 80640, 0, -169344, 0, 241920, 181440, 40320, 0, -544320, 0, 1814400, 0, -2540160, 0, 2721600, 1814400, 362880

Offset: 0

Author

Wouter Meeussen, Mar 11 2003

Keywords

• easy
• sign
• tabl

Comments

The harmonic numbers as used here are defined: HarmonicNumber(n, r) = Zeta(r) - HurwitzZeta(r, n + 1). - Peter Luschny, Mar 21 2024

Examples

			The triangle t(n, m) begins:
n\m  0    1    2      3    4      5     6    7 ...
0:   1
1:   1    1
2:   1    3    2
3:   0    6   12      6
4:  -4    0   40     60   24
5:   0  -60    0    300  360    120
6: 120    0 -840      0 2520   2520   720
7:   0 3360    0 -11760    0  23520 20160 5040
...
Row n=8: -12096    0 80640      0 -169344 0 241920 181440 40320;
Row n=9: 0 -544320 0 1814400 0 -2540160 0 2721600 1814400 362880;
Row n=10: 3024000 0 -19958400 0 39916800 0 -39916800 0 33264000 19958400 3628800.
... Reformatted and extended. - _Wolfdieter Lang_, Feb 04 2016
		

References

• J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, pp. 370 - 371.

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened
• Wikipedia, Faulhaber's formula

Crossrefs

Cf. A264388, A264389, A103438.

Programs

Maple

RowPoly := n -> local j; n!*add(binomial(n + 1, j) * bernoulli(j, 1) * x^(n - j), j = 0..n): seq(lprint(seq(coeff(RowPoly(n), x, k), k = 0..n)), n = 0..8);
# Peter Luschny, Mar 21 2024

Mathematica

Table[(n+1)! CoefficientList[Sum[k^n, {k, 0, m}]/m, m], {n,1,12}] and for n=0: 1.
a = Join[{{1}}, Table[CoefficientList[Expand[n!*(BernoulliB[n + 1, x + 1] - BernoulliB[n + 1])/x], x], {n, 1, 10}]] Flatten[a] (* Roger L. Bagula and N. J. A. Sloane, Feb 18 2008 *)
T[n_, k_] := Coefficient[ 1/x Integrate[ BernoulliB[n, x + 1], x], x, k]; (* Michael Somos, Aug 18 2018 *)

Formula

Row sums are (n + 1)!, last element in row n is n!

Alternative description using Bernoulli polynomials: Let p[x,n]=Sum[k^n,{k,1,x}]; 1/x /. NSolve[p[x,n]-Zeta[n]==0,x] where n>=2. Then t(n,m) = CoefficientList[Expand[n!*(BernoulliB[n + 1, x + 1] - BernoulliB[n + 1])/x], x]. - Roger L. Bagula and N. J. A. Sloane, Feb 18 2008

From Wolfdieter Lang, Feb 04 2016: (Start)

The row polynomials R(n, x) = (n+1)!*F(n, x)/x with F(n,x) = (Sum_{k=1..m} k^n)|{m=x} satisfy the recurrence R(n, x) = n!*(((x + 1)^(n+1) - 1)/x - Sum{k=0..n-1} (binomial(n+1, k)*R(k, x)/(k+1)!)), n >= 1, and R(0, x) = 1. See the Silverman reference, pp. 370 - 371, for F(n, x).

t(n, m) = [x^m] ((Bernoulli(n+1, x+1) - Bernoulli(n+1, 1))/x). See a comment above. For these Bernoulli polynomials see A264388 and A264389. (End)

t(n, m) = t(n-1, m-1) * n/(m+1). - Michael Somos, Aug 18 2018

T(n, k) = [x^k] n!*Sum_{j=0..n} binomial(n+1, j)*Bernoulli(j, 1)*x^(n - j). - Peter Luschny, Mar 21 2024