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A377506 a(n) is the nearest integer to 1/gamma(x_n), where x_n is the n-th extrema of gamma(x).

%I A377506 #33 Jan 12 2025 17:29:24
%S A377506 1,0,0,-1,4,-19,107,-716,5498,-47789,463517,-4962289,58115593,
%T A377506 -739030560,10140362326,-149320366368,2348685116841,-39299792354491,
%U A377506 697018000148170,-13061370974841665,257854085426453001,-5349016057902489052,116324040001711961903,-2646269955793816322943,62852365790563502907461
%N A377506 a(n) is the nearest integer to 1/gamma(x_n), where x_n is the n-th extrema of gamma(x).
%C A377506 a(n) approximately equals round(1/gamma(-n+ arctan(pi/log(n)) / pi)).
%C A377506 This sequence shows how rapidly the extrema of the gamma function approach the real axis on the negative side.
%H A377506 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gamma_function#Minima_and_maxima">Gamma Extrema</a>
%H A377506 Wikipedia, <a href="https://en.wikipedia.org/wiki/Digamma_function#Roots_of_the_digamma_function">Digamma Roots</a>
%e A377506 a(0) = round(1/Γ(x_0)) = round(1/Γ(1.4616321449683622)) = round(1/0.8856031944108887) = 1
%e A377506 a(1) = round(1/Γ(x_1)) = round(1/Γ(-0.5040830082644554)) = round(1/-3.544643611155005) = 0
%e A377506 where x_m is the m-th extrema of gamma(x) or equivalently the m-th root of digamma(x).
%t A377506 a[n_] := Module[{root},
%t A377506   root = FindRoot[PolyGamma[0, x] == 0, {x, -n + 10^-10, -n + 0.5}, WorkingPrecision -> 100];
%t A377506   Round[1/Gamma[x /. root]]
%t A377506 ]
%t A377506 Table[a[n], {n, 0, 24}]
%o A377506 (Python)
%o A377506 from mpmath import findroot, digamma, mp, iv
%o A377506 mp.dps = iv.dps = 30
%o A377506 def generate_sequence(n):
%o A377506     seq, gamma_extrema = [], 1.4616321449683622
%o A377506     for i in range(n):
%o A377506         gamma_extrema = findroot(lambda x: digamma(x+1), x0=gamma_extrema-1)
%o A377506         reci_gamma = iv.rgamma(gamma_extrema+1)
%o A377506         gamma_extrema = -mp.fabs(gamma_extrema)
%o A377506         assert -i-1 < -mp.fabs(gamma_extrema) < -i-0.4
%o A377506         nint_reci = mp.nint(reci_gamma.a)
%o A377506         assert nint_reci == mp.nint(reci_gamma.b)
%o A377506         seq.append(int(nint_reci))
%o A377506     return seq
%o A377506 A377506 = generate_sequence(n=25)
%Y A377506 Cf. A374856.
%K A377506 sign
%O A377506 1,5
%A A377506 _Jwalin Bhatt_, Oct 30 2024