A180085 Eulerian polynomials at nonpositive integers, A_{n}(-n).

1, 1, -1, -2, 69, -1104, 11395, 189232, -21769335, 1156775680, -41290278129, -136576564992, 234678001445965, -32256618406068224, 3018646161081366075, -158289126522080405504, -15471427638848015017455, 6998210972374723086974976, -1487059744246923349187974457, 223233959091253143036239872000

Offset

0,4

Comments

Define the Eulerian numbers A(n,k) as the number of permutations of {1,2,..,n} with k ascents and the Eulerian polynomials A_{0}(x) = 1; A_{n}(x) = sum_{k=0..n-1} A(n,k) x^k for n > 0. Then a(n) = A_{n}(-n) are the values of the Eulerian polynomials for n = 0,-1,-2,-3,...

Links

Table of n, a(n) for n=0..19.

OEIS Wiki, Eulerian polynomials.

Formula

a(n) = -(n+1)^(n+1)*Li_{-n}(n)/n, where Li_{n}(z) denotes the polylogarithm. For n = 0, interpret it as a limit for continuous n -> 0, that gives a(0) = 1. - Vladimir Reshetnikov, Oct 15 2016

a(n) = n! * [x^n] (n + 1) / (n + exp(-(n + 1)*x)). - Ilya Gutkovskiy, Jun 28 2020

Maple

c := proc(n, m) local k; add((-1)^k*binomial(n+1, k)*(m+1-k)^n, k=0..m) end:
a := proc(n) local k; `if`(n=0, 1, add(c(n, k)*(-n)^k, k=0..n-1)) end:

Mathematica

a[0] = 1; a[n_] := -(n + 1)^(n + 1) PolyLog[-n, -n]/n; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2016 *)

Prog

(PARI) a(n) = if (n==0, 1, -(n + 1)^(n + 1)*polylog(-n, -n)/n); \\ Michel Marcus, May 30 2018

Crossrefs

Cf. A122778

Sequence in context: A362291 A364854 A041577 * A321244 A224506 A085915

Adjacent sequences: A180082 A180083 A180084 * A180086 A180087 A180088

Keyword

sign

Author

Peter Luschny, Aug 12 2010

Extensions

More terms from Michel Marcus, May 30 2018

Status

approved