OFFSET
0,4
COMMENTS
The Bernoulli numbers with B(1) = 1/2 can be represented as the weighted sum of Eulerian numbers of second order, where we use the definition as given by Graham et al., Eulerian2(n, k) = A201637(n, k). For n >= 1 we have
B_(n) = (1/2)*Sum_{k=0..n} (-1)^k*Eulerian2(n, k) / binomial(2*n - 1, k).
Although this representation looks classical it was apparently first proved by Majer in 2010; later Fu and recently O'Sullivan gave an alternative proof (see links).
An analogous representation based on the Eulerian numbers of first order is given in A356545.
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270. (Since the thirty-fourth printing, Jan. 2022, with B(1) = 1/2.)
LINKS
Amy M. Fu, Some Identities Related to the Second-Order Eulerian Numbers, arXiv:2104.09316 [math.CO], Apr. 2021.
Peter Luschny, How are the Eulerian numbers of the first-order related to the Eulerian numbers of the second-order?, MathOverflow, Feb. 2021.
Pietro Majer, Expressions involving Eulerian numbers of the second kind, MathOverflow, Nov 2010.
G. Rzadkowski, M. Urlinska, A Generalization of the Eulerian Numbers, arXiv:1612.06635 [math.CO], 2016
Cormac O'Sullivan, Stirling's approximation and a hidden link between two of Ramanujan's approximations, arXiv:2208.02898 [math.NT], Aug. 2022.
FORMULA
Let p_n(x) = Sum_{k=0..n} Eulerian2(n, k)*k!*(2*n - k - 1)! * (-x)^k.
T(n, k) = [x^k] p_n(x).
T(n, k) = (-1)^k*Eulerian2(n, k)*k!*(2*n - k - 1)!.
EXAMPLE
The triangle T(n, k) of the coefficients, sorted in ascending order, starts:
[0] 1;
[1] 1, 0;
[2] 6, -4, 0;
[3] 120, -192, 72, 0;
[4] 5040, -15840, 13920, -3456, 0;
[5] 362880, -2096640, 3306240, -1918080, 345600, 0;
[6] 39916800, -413683200, 1053803520, -1064448000, 448519680, -62208000, 0;
MAPLE
E2 := proc(n, k) combinat:-eulerian2(n, k) end:
p := (n, x) -> `if`(n = 0, 1, add(E2(n, k)*k!*(2*n - k - 1)!*(-x)^k, k = 0..n)):
seq(print([n], seq(coeff(p(n, x), x, k), k = 0..n)), n = 0..7);
seq(`if`(n = 0, 1, p(n, 1)/(2*(2*n-1)!)), n = 0..14); # check Bernoulli numbers
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Aug 12 2022
STATUS
approved