A335951 Triangle read by rows. The numerators of the coefficients of the Faulhaber polynomials. T(n,k) for n >= 0 and 0 <= k <= n.

1, 0, 1, 0, 0, 1, 0, 0, -1, 4, 0, 0, 1, -4, 6, 0, 0, -3, 12, -20, 16, 0, 0, 5, -20, 34, -32, 16, 0, 0, -691, 2764, -4720, 4592, -2800, 960, 0, 0, 105, -420, 718, -704, 448, -192, 48, 0, 0, -10851, 43404, -74220, 72912, -46880, 21120, -6720, 1280

Offset: 0

Author

Peter Luschny, Jul 16 2020

Keywords

• sign
• tabl
• frac

Comments

There are many versions of Faulhaber's triangle: search the OEIS for his name.

Faulhaber's claim (in 1631) is: S_{2*m-1} = 1^(2*m-1) + 2^(2*m-1) + ... + n^(2*m-1) = F_m((n^2+2)/2). The first proof was given by Jacobi in 1834.

For the Faulhaber numbers see A354042 and A354043.

Examples

			The first few polynomials are:
  [0] 1;
  [1] x;
  [2] x^2;
  [3] (4*x - 1)*x^2*(1/3);
  [4] (6*x^2 - 4*x + 1)*x^2*(1/3);
  [5] (16*x^3 - 20*x^2 + 12*x - 3)*x^2*(1/5);
  [6] (16*x^4 - 32*x^3 + 34*x^2 - 20*x + 5)*x^2*(1/3);
  [7] (960*x^5 - 2800*x^4 + 4592*x^3 - 4720*x^2 + 2764*x - 691)*x^2*(1/105);
  [8] (48*x^6 - 192*x^5 + 448*x^4 - 704*x^3 + 718*x^2 - 420*x + 105)*x^2*(1/3);
  [9] (1280*x^7-6720*x^6+21120*x^5-46880*x^4+72912*x^3-74220*x^2+43404*x-10851)*x^2*(1/45);
Triangle starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 0,  1;
  [3] 0, 0, -1,     4;
  [4] 0, 0,  1,    -4,      6;
  [5] 0, 0, -3,     12,    -20,    16;
  [6] 0, 0,  5,    -20,     34,   -32,     16;
  [7] 0, 0, -691,   2764,  -4720,  4592,  -2800,  960;
  [8] 0, 0,  105,  -420,    718,  -704,    448,  -192,    48;
  [9] 0, 0, -10851, 43404, -74220, 72912, -46880, 21120, -6720, 1280;
		

References

• Johann Faulhaber, Academia Algebra. Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden. Johann Ulrich Schönigs, Augsburg, 1631.

Links

• C. G. J. Jacobi, De usu legitimo formulae summatoriae Maclaurinianae, J. Reine Angew. Math., 12 (1834), 263-272.
• Donald E. Knuth, Johann Faulhaber and sums of powers, arXiv:math/9207222 [math.CA], 1992; Math. Comp. 61 (1993), no. 203, 277-294.
• Peter Luschny, Illustrating the Faulhaber polynomials for n = 1..7.

Crossrefs

Cf. A335952 (polynomial denominators), A000012 (row sums of the polynomial coefficients).

Other representations of the Faulhaber polynomials include A093556/A093557, A162298/A162299, A220962/A220963.

Cf. A354042 (Faulhaber numbers), A354043.

Programs

Maple

FaulhaberPolynomial := proc(n) if n = 0 then return 1 fi;
expand((bernoulli(2*n, x+1) - bernoulli(2*n,1))/(2*n));
sort(simplify(expand(subs(x = (sqrt(8*x+1)-1)/2, %))), [x], ascending) end:
Trow := n -> seq(coeff(numer(FaulhaberPolynomial(n)), x, k), k=0..n):
seq(print(Trow(n)), n=0..9);

Python

from math import lcm
from itertools import count, islice
from sympy import simplify,sqrt,bernoulli
from sympy.abc import x
def A335951_T(n,k):
    z = simplify((bernoulli(2*n,(sqrt(8*x+1)+1)/2)-bernoulli(2*n,1))/(2*n)).as_poly().all_coeffs()
    return z[n-k]*lcm(*(d.q for d in z))
def A335951_gen(): # generator of terms
    yield from (A335951_T(n,k) for n in count(0) for k in range(n+1))
A335951_list = list(islice(A335951_gen(),20)) # Chai Wah Wu, May 16 2022

SageMath

def A335951Row(n):
    R. = PolynomialRing(QQ)
    if n == 0: return [1]
    b = expand((bernoulli_polynomial(x + 1, 2*n) -
                bernoulli_polynomial(1, 2*n))/(2*n))
    s = expand(b.subs(x = (sqrt(8*x+1)-1)/2))
    return numerator(s).list()
for n in range(10): print(A335951Row(n)) # Peter Luschny, May 17 2022

Formula

Let F_n(x) be the polynomial after substituting (sqrt(8*x + 1) - 1)/2 for x in b_n(x), where b_n(x) = (Bernoulli_{2*n}(x+1) - Bernoulli_{2*n}(1))/(2*n).

F_n(1) = 1 for all n >= 0.

T(n, k) = numerator([x^k] F_n(x)).