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A177885 a(n) = (1-n)^(n-1). 17

%I #70 Apr 17 2023 06:35:04

%S 1,1,-1,4,-27,256,-3125,46656,-823543,16777216,-387420489,10000000000,

%T -285311670611,8916100448256,-302875106592253,11112006825558016,

%U -437893890380859375,18446744073709551616,-827240261886336764177

%N a(n) = (1-n)^(n-1).

%C A signed version of A000312.

%C 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]

%H Vincenzo Librandi, <a href="/A177885/b177885.txt">Table of n, a(n) for n = 0..140</a>

%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.

%H A. LeClair, <a href="http://arxiv.org/abs/1305.2613">An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N</a>, arXiv:1305.2613 [math-ph], 2013.

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

%F 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

%F E.g.f.: 1 + Series_Reversion( (1+x)*log(1+x) ). - _Paul D. Hanna_, Aug 24 2016

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

%F a(n) ~ (-1)^(n+1) * exp(-1) * n^(n-1). - _Vaclav Kotesovec_, Sep 22 2016

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1, k-1)*n^(n-k), for n >= 1 and a(0) = 1, that is, Sum_{k=0..n}*A137452(n, k), for n >= 0. - _Wolfdieter Lang_, Apr 11 2023

%e From _Paul D. Hanna_, Aug 24 2016: (Start)

%e 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! +...

%e Related series.

%e 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)

%t Join[{1,1}, Table[(1-n)^(n-1), {n, 2, 20}]] (* _Harvey P. Dale_, Aug 10 2012 *)

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

%o (Magma) [(1-n)^(n-1): n in [0..30]]; // _Vincenzo Librandi_, May 15 2011

%o (PARI) a(n)=(1-n)^(n-1) \\ _Charles R Greathouse IV_, May 15 2013

%o (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)}

%o for(n=0,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Aug 24 2016

%Y Cf. A000312, A137452 (row sums).

%K sign,easy

%O 0,4

%A _Vladimir Kruchinin_, Dec 28 2010

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)