A354042 -id:A354042 - cp's OEIS frontend

A354043 Table read by rows: T(n, k) = (-1)^(n-k)*F(n, k)/k!, where F are the Faulhaber numbers A354042.

1, 0, 1, 0, 1, 1, 0, 4, 4, 1, 0, 36, 36, 10, 1, 0, 600, 600, 170, 20, 1, 0, 16584, 16584, 4720, 574, 35, 1, 0, 705600, 705600, 201040, 24640, 1568, 56, 1, 0, 43751232, 43751232, 12468960, 1531152, 98448, 3696, 84, 1, 0, 3790108800, 3790108800, 1080240480, 132713280, 8554896, 325152, 7812, 120, 1

Offset: 0

Peter Luschny, May 17 2022

I. Gessel and X. Viennot give two combinatorial interpretations for the Faulhaber numbers (see link). We quote their theorems 32 an 33, using our notation:
Theorem: T(n, k) is the number of row-strict tableaux of shape (n - k + 2, n - k + 1, ..., 2) - (n - k - 1, n - k - 2, ..., 0) with positive integer entries in which the largest entry in row i is at most n + 2 - i.
Theorem: T(n, k) is the number of sequences a_{1} a_{2} ... a_{3n-3k} of positive integers satisfying a_{3i-2} < a_{3i-1} < a_{3i}, a_{3i-1} >= a_{3i+1}, a_{3i} >= a_{3i+2}, and a_{3i} <= k + i + 1 for all i.

			Table starts:
  [0] 1;
  [1] 0,        1;
  [2] 0,        1,        1;
  [3] 0,        4,        4,        1;
  [4] 0,       36,       36,       10,       1;
  [5] 0,      600,      600,      170,      20,     1;
  [6] 0,    16584,    16584,     4720,     574,    35,    1;
  [7] 0,   705600,   705600,   201040,   24640,  1568,   56,  1;
  [8] 0, 43751232, 43751232, 12468960, 1531152, 98448, 3696, 84, 1;

I. M. Gessel and X. G. Viennot, Determinants, Paths, and Plane Partitions, 1989 preprint.

Cf. A354042, A354045 (row sums).

T := (n, k) -> ifelse(n = 0, 1, (-1)^n*((n + 1)!/k!)*add(binomial(2*k - 2*j, k + 1)*binomial(2*n + 1, 2*j + 1)*bernoulli(2*n - 2*j) / (j - k), j = 0..(k-1)/2)): for n from 0 to 8 do seq(T(n, k), k = 0..n) od;

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

Original entry on oeis.org

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

Peter Luschny, Jul 16 2020

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.

			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;

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

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.

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.

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);

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

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

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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A354043 Table read by rows: T(n, k) = (-1)^(n-k)*F(n, k)/k!, where F are the Faulhaber numbers A354042.

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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.

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A354045 Row sums of A354043.

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