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%I #30 Mar 21 2024 09:21:05
%S 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,
%T 2520,2520,720,0,3360,0,-11760,0,23520,20160,5040,-12096,0,80640,0,
%U -169344,0,241920,181440,40320,0,-544320,0,1814400,0,-2540160,0,2721600,1814400,362880
%N Triangle read by rows: n-th row gives expansion of the series for HarmonicNumber(n, -r).
%C The harmonic numbers as used here are defined: HarmonicNumber(n, r) = Zeta(r) - HurwitzZeta(r, n + 1). - _Peter Luschny_, Mar 21 2024
%D J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, pp. 370 - 371.
%H G. C. Greubel, <a href="/A080779/b080779.txt">Rows n=0..100 of triangle, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Faulhaber%27s_formula">Faulhaber's formula</a>
%F Row sums are (n + 1)!, last element in row n is n!
%F 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
%F From _Wolfdieter Lang_, Feb 04 2016: (Start)
%F 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).
%F 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)
%F t(n, m) = t(n-1, m-1) * n/(m+1). - _Michael Somos_, Aug 18 2018
%F 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
%e The triangle t(n, m) begins:
%e n\m 0 1 2 3 4 5 6 7 ...
%e 0: 1
%e 1: 1 1
%e 2: 1 3 2
%e 3: 0 6 12 6
%e 4: -4 0 40 60 24
%e 5: 0 -60 0 300 360 120
%e 6: 120 0 -840 0 2520 2520 720
%e 7: 0 3360 0 -11760 0 23520 20160 5040
%e ...
%e Row n=8: -12096 0 80640 0 -169344 0 241920 181440 40320;
%e Row n=9: 0 -544320 0 1814400 0 -2540160 0 2721600 1814400 362880;
%e Row n=10: 3024000 0 -19958400 0 39916800 0 -39916800 0 33264000 19958400 3628800.
%e ... Reformatted and extended. - _Wolfdieter Lang_, Feb 04 2016
%p 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);
%p # _Peter Luschny_, Mar 21 2024
%t Table[(n+1)! CoefficientList[Sum[k^n, {k, 0, m}]/m, m], {n,1,12}] and for n=0: 1.
%t 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 T[n_, k_] := Coefficient[ 1/x Integrate[ BernoulliB[n, x + 1], x], x, k]; (* _Michael Somos_, Aug 18 2018 *)
%Y Cf. A264388, A264389, A103438.
%K easy,sign,tabl
%O 0,5
%A _Wouter Meeussen_, Mar 11 2003
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