OFFSET
4,1
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.
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.
Donald 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.
EXAMPLE
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).
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]
CROSSREFS
KEYWORD
sign
AUTHOR
Roman Witula, Dec 11 2014
STATUS
approved