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%I #24 May 21 2023 22:23:02
%S 1,-1,-1,1,1,1,-1,-1,-1,1,1,1,-1,-1,-1,-1,1,1,1,-1,-1,-1,-1,1,1,1,1,
%T -1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1,1,-1,-1,-1,-1,1,1,1,1,-1,-1,-1,
%U -1,1,1,1,1,1,-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,-1,1,1
%N Sign of n-th coefficient of power series for 1/Gamma(1-x) where Gamma is the Gamma function.
%C Computing this sequence was discussed on the seqfan mailing list in May 2023. Typical floating-point precision is not sufficient to compute more than a few terms of this sequence. Therefore, either a high precision needs to be set; e.g, a precision of round(n*log(n)/Pi) [see formula (3.21) in Fekih-Ahmed], or the computation must be done with an arbitrary precision method (e.g. interval arithmetic, FLINT, or the Java below). - _Sean A. Irvine_, May 21 2023
%H G. C. Greubel, <a href="/A063747/b063747.txt">Table of n, a(n) for n = 0..2500</a>
%H Lazhar Fekih-Ahmed, <a href="https://hal.science/hal-01029331v1/document">On the Power Series Expansion of the Reciprocal Gamma Function</a>, 2014.
%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a063/A063747.java">Java program</a> (github)
%e 1/Gamma(1-x) = 1 - gamma*x - 0.6558...*x^2 + 0.042...*x^3 + 0.1665...*x^4 + 0.0421...*x^5 - 0.0096..*x^6 +.... hence sequence begins 1,-1,-1,1,1,1,-1,...
%e [x^46] 1/Gamma(1-x) = -4.445829736550756882101590352124643637401436685748718288670...*10^-39, from which a(46)=-1. _Sean A. Irvine_, May 21 2023
%t precis = 200;
%t g[1, k_] := g[1, k] = EulerGamma^k/k!;
%t g[m_, k_] := g[m, k] = Zeta[m]^k/(k! m^k) // N[#, precis]&;
%t f[0, 0] = 1; f[n_ /; n > 0, 0] = 0;
%t f[n_, m_] := f[n, m] = Sum[g[m, k]*f[n - k m, m - 1],
%t {k, 0, n/m}] // N[#, precis]&;
%t c[i_] := f[i, i]; b[0] = 1;
%t b[i_] := b[i] = -Sum[c[j]*b[i - j], {j, 1, i}];
%t a[n_] := Sign[b[n]];
%t Table[a[n], {n, 0, 74}] (* _Jean-François Alcover_, May 20 2023, after _Brendan McKay_ on seqfan *)
%o (PARI) a(n)=sign(polcoeff(1/(gamma(1-x +O(x^(n+1)))),n))
%o (PARI) my(x='x+O('x^100)); apply(sign, Vec(1/(gamma(1-x)))) \\ _Michel Marcus_, May 19 2023
%K sign
%O 0,1
%A _Benoit Cloitre_, Jan 16 2004
%E a(46) onward corrected by _Sean A. Irvine_, May 19 2023
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