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A177885 a(n) = (1-n)^(n-1). 8
1, 1, -1, 4, -27, 256, -3125, 46656, -823543, 16777216, -387420489, 10000000000, -285311670611, 8916100448256, -302875106592253, 11112006825558016, -437893890380859375, 18446744073709551616, -827240261886336764177 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A signed version of A000312.

LeClair gives an approximation z(n) for the location of the n-th nontrivial zero of the Riemann zeta function on the critical line, which can be expressed in terms of the exponential generating function of this sequence A(x) = x/LambertW(x) as follows: z(n) = 1/2 + 2*Pi*exp(1)*A((n - 11/8)/exp(1))*i. For example, working to 1 decimal place, z(1) = 1/2 + 14.5*i (the first nontrivial zero is at 1/2 + 14.1*i), z(10) = 1/2 + 50.2*i (the tenth nontrivial zero is at 1/2 + 49.8*i) and z(100) = 1/2 + 236*i (the hundredth nontrivial zero is at 1/2 + 236.5*i). [Peter Bala, Jun 12 2013]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..140

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582

A. LeClair, An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N, arXiv:1305.2613v4[math-ph]

FORMULA

E.g.f. satisfies A(x) = exp(x/A(x)).

E.g.f. A(x) = x/LambertW(x) = exp(LambertW(x)) = 1 + x - x^2/2! + 4*x^3/3! - 27*x^4/4! + .... - Peter Bala, Jun 12 2013

E.g.f.: 1 + Series_Reversion( (1+x)*log(1+x) ). - Paul D. Hanna, Aug 24 2016

E.g.f.: 1 + Series_Reversion( x + Sum_{n>=2} (-x)^n/(n*(n-1)) ). - Paul D. Hanna, Aug 24 2016

a(n) ~ (-1)^(n+1) * exp(-1) * n^(n-1). - Vaclav Kotesovec, Sep 22 2016

EXAMPLE

From Paul D. Hanna, Aug 24 2016: (Start)

E.g.f.: A(x) = 1 + x - x^2/2! + 4*x^3/3! - 27*x^4/4! + 256*x^5/5! - 3125*x^6/6! + 46656*x^7/7! - 823543*x^8/8! +...+ (1-n)^(n-1)*x^n/n! +...

Related series.

Series_Reversion(A(x) - 1) = x + x^2/2 - x^3/6 + x^4/12 - x^5/20 + x^6/30 - x^7/42 + x^8/56 - x^9/72 + x^10/90 +...+ (-x)^n/(n*(n-1)) +... (End)

MATHEMATICA

Join[{1, 1}, Table[(1-n)^(n-1), {n, 2, 20}]] (* Harvey P. Dale, Aug 10 2012 *)

nn = 18; Range[0, nn]! CoefficientList[ Series[ Exp[ ProductLog[ x]], {x, 0, nn}], x] (* Robert G. Wilson v, Aug 23 2012 *)

PROG

(MAGMA) [(1-n)^(n-1): n in [0..30]]; // Vincenzo Librandi, May 15 2011

(PARI) a(n)=(1-n)^(n-1) \\ Charles R Greathouse IV, May 15 2013

(PARI) {a(n) = my(A = 1 + serreverse( x + sum(m=2, n+2, (-x)^m/(m*(m-1)) +x^2*O(x^n)))); n!*polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 24 2016

CROSSREFS

Cf. A000312.

Sequence in context: A070271 A245414 A000312 * A086783 A050764 A240582

Adjacent sequences:  A177882 A177883 A177884 * A177886 A177887 A177888

KEYWORD

sign,easy

AUTHOR

Vladimir Kruchinin, Dec 28 2010

STATUS

approved

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Last modified February 24 07:50 EST 2018. Contains 299599 sequences. (Running on oeis4.)