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A335951
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Triangle read by rows. The numerators of the coefficients of the Faulhaber polynomials. T(n,k) for n >= 0 and 0 <= k <= n.
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3
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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
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OFFSET
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0,10
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COMMENTS
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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.
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REFERENCES
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Johann Faulhaber, Academia Algebra. Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden. Johann Ulrich Schönigs, Augsburg, 1631.
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LINKS
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FORMULA
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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)).
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EXAMPLE
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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);
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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;
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MAPLE
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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);
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PROG
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(Python)
from math import lcm
from itertools import count, islice
from sympy import simplify, sqrt, bernoulli
from sympy.abc import x
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))
(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
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CROSSREFS
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Cf. A335952 (polynomial denominators), A000012 (row sums of the polynomial coefficients).
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KEYWORD
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AUTHOR
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STATUS
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approved
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