A251926 The Faulhaber-Knuth a(0,n) sequence.

2, 1, 1, 1, 1, 0, 0, 1, 37, -60, -5, 37, 174, -955, -10545, 38610, 176297, -322740, -205420, 4512655, 56820585, -104019264, -25907081, 94854194, 1141847218, -2090335775, -414239903275, 6066664425833, 85621405759989, -156743813184120, -4337631088920, 47644406040193, 1265208493396175131, -2316168508680582540, -192288633159406495

Offset: 4

Author

Roman Witula, Dec 11 2014

Keywords

• sign

Comments

a(n) is equal to the remainder when dividing the polynomial T_n(x) by x^2 + x - 1. T_n(x) (in Z[x]) is the positive integer multiplicity of the modified Faulhaber polynomial T*_n(x), coefficients of which have GCD equal to 1. We have T*_n(x) = S(n;x)/x^2(x+1)^2 if n is odd, and T*_n(x) = S(n;x)/x(x+1)(2x+1) if n is even, n >= 4, where S(n;x) denotes the n-th Faulhaber polynomial, i.e., S(n;x) = 1/(n+1) sum{taken over i=0,1,...,n} Bin(n+1,i)Bern(i)x^(n+1-i), and Bern(i) denotes the i-th Bernoulli number with Bern(1)=1/2.

We note that every T_n(x) is a polynomial in the variable (x^2 + x - 1), for example T_7(x) = 3(x^2 + x - 1)^2 + 2(x^2 + x - 1) + 1. Furthermore, every T_n(x) is a polynomial in (x^2 + x + a) for each complex a. But only for a = -1 is the element a(n) also equal to the remainder when dividing S(n;x) by x^2 + x + a if n is odd and S(n;x)/(2x+1) by x^2 + x + a if n is even.

Examples

			We have: T_4(x) = 3x^2 + 3x - 1, T_4(x) - T_5(x) = x^2 + x, T_6(x) - T_7(x) = x^2 + x - 1, T_9(x) = (x^2 + x - 1)(2x^4 + 4x^3 - x^2 - 3x + 3) and T_15(x) - T_12(x) is divisible by (x^2 + x - 1), which implies a(0)=2, a(1)=1, a(2)=a(3), a(5)=0 and a(8)=a(11).
		

Links

• Edyta Hetmaniok, Piotr Lorenc, Mariusz Pleszczyński, and Roman Wituła, Iterated integrals of polynomials, Applied Mathematics and Computation, Volume 249, 15 December 2014, Pages 389-398.
• D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comput. 203 (1993), 277-294.
• Piotr Lorenc, Jakub Jan Ludew, Mariusz Pleszczyński, Alicja Samulewicz, and Roman Wituła, Iterated integrals of Faulhaber polynomials and some properties of their roots, 2018.

Crossrefs

Cf. A093556, A093557.

Programs

Mathematica

coeffFaulh[n_] := Module[{t, tab = {}, s, p, x},
  If[n < 4, Return["Give n greater than 3."]];
  t = Table[1, {n + 2}];
  Do[t[[i + 1]] = BernoulliB[i], {i, 1, n + 1}];
  t[[2]] = 1/2;
  s[m_, x_] := (Sum[Binomial[m + 1, i]t[[ i + 1]] x^(m + 1 - i),{i,0,m}])/(m + 1);
  Do[If[Mod[i, 2] == 0,
    p = PolynomialRemainder[FactorList[Factor[s[i, x]] (i + 1)/(x (x + 1) (2 x + 1))][[2,1]], -1 + x + x^2, x],
    p = PolynomialRemainder[FactorList[Factor[s[i, x]] (i + 1)/(x^2 (x + 1)^2)][[2,1]], -1 + x + x^2, x]];
   tab = Append[tab, p], {i, 4, n}];
  tab]