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A335951
Triangle read by rows. The numerators of the coefficients of the Faulhaber polynomials. T(n,k) for n >= 0 and 0 <= k <= n.
3
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
0,10
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.
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.
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)).
EXAMPLE
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;
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);
PROG
(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.<x> = 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
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.
Sequence in context: A245965 A078669 A229655 * A254156 A344386 A046783
KEYWORD
sign,tabl,frac
AUTHOR
Peter Luschny, Jul 16 2020
STATUS
approved