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

4,1

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

Table of n, a(n) for n=4..38.

D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comput. 203 (1993), 277-294.

Piotr Lorenc, Jakub Jan Ludew, Mariusz Pleszczyński, Alicja Samulewicz, Roman Wituła, Iterated integrals of Faulhaber polynomials and some properties of their roots, 2018.

P. Lorenc, M. Pleszczyński, R. Witula, Iterated integrals of polynomials, Applied Mathematics and Computation, Volume 249, 15 December 2014, Pages 389-398.

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

Cf. A093556, A093557.

Sequence in context: A339366 A016397 A238450 * A335504 A037908 A116663

Adjacent sequences: A251923 A251924 A251925 * A251927 A251928 A251929

Keyword

sign

Author

Roman Witula, Dec 11 2014

Status

approved