A356547 Triangle read by rows. T(n, k) are the coefficients of polynomials p_n(x) based on the Eulerian numbers of second order representing the Bernoulli numbers as B_n = p_n(1) / (2*(2*n - 1)!).

1, 1, 0, 6, -4, 0, 120, -192, 72, 0, 5040, -15840, 13920, -3456, 0, 362880, -2096640, 3306240, -1918080, 345600, 0, 39916800, -413683200, 1053803520, -1064448000, 448519680, -62208000, 0, 6227020800, -114960384000, 447866496000, -699342336000, 506348236800, -164428185600, 18289152000, 0

Offset: 0

Author

Peter Luschny, Aug 12 2022

Keywords

• sign
• tabl

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.

Examples

			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;
		

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.

Crossrefs

Cf. A201637 (Eulerian number 2nd order), A164555(n)/A027642(n) (Bernoulli numbers with B(1) = 1/2).

Cf. A356545.

Programs

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

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)!.