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A180085 Eulerian polynomials at nonpositive integers, A_{n}(-n). 0
1, 1, -1, -2, 69, -1104, 11395, 189232, -21769335, 1156775680, -41290278129, -136576564992, 234678001445965, -32256618406068224, 3018646161081366075, -158289126522080405504, -15471427638848015017455 (list; graph; refs; listen; history; text; internal format)
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..16.

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

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 *)

CROSSREFS

Cf. A122778

Sequence in context: A264457 A246744 A041577 * A224506 A085915 A285217

Adjacent sequences:  A180082 A180083 A180084 * A180086 A180087 A180088

KEYWORD

sign

AUTHOR

Peter Luschny, Aug 12 2010

STATUS

approved

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Last modified May 23 08:03 EDT 2017. Contains 286909 sequences.