A377506 - OEIS

%I #33 Jan 12 2025 17:29:24

%S 1,0,0,-1,4,-19,107,-716,5498,-47789,463517,-4962289,58115593,

%T -739030560,10140362326,-149320366368,2348685116841,-39299792354491,

%U 697018000148170,-13061370974841665,257854085426453001,-5349016057902489052,116324040001711961903,-2646269955793816322943,62852365790563502907461

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

%C a(n) approximately equals round(1/gamma(-n+ arctan(pi/log(n)) / pi)).

%C This sequence shows how rapidly the extrema of the gamma function approach the real axis on the negative side.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gamma_function#Minima_and_maxima">Gamma Extrema</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Digamma_function#Roots_of_the_digamma_function">Digamma Roots</a>

%e a(0) = round(1/Γ(x_0)) = round(1/Γ(1.4616321449683622)) = round(1/0.8856031944108887) = 1

%e a(1) = round(1/Γ(x_1)) = round(1/Γ(-0.5040830082644554)) = round(1/-3.544643611155005) = 0

%e where x_m is the m-th extrema of gamma(x) or equivalently the m-th root of digamma(x).

%t a[n_] := Module[{root},

%t   root = FindRoot[PolyGamma[0, x] == 0, {x, -n + 10^-10, -n + 0.5}, WorkingPrecision -> 100];

%t   Round[1/Gamma[x /. root]]

%t ]

%t Table[a[n], {n, 0, 24}]

%o (Python)

%o from mpmath import findroot, digamma, mp, iv

%o mp.dps = iv.dps = 30

%o def generate_sequence(n):

%o     seq, gamma_extrema = [], 1.4616321449683622

%o     for i in range(n):

%o         gamma_extrema = findroot(lambda x: digamma(x+1), x0=gamma_extrema-1)

%o         reci_gamma = iv.rgamma(gamma_extrema+1)

%o         gamma_extrema = -mp.fabs(gamma_extrema)

%o         assert -i-1 < -mp.fabs(gamma_extrema) < -i-0.4

%o         nint_reci = mp.nint(reci_gamma.a)

%o         assert nint_reci == mp.nint(reci_gamma.b)

%o         seq.append(int(nint_reci))

%o     return seq

%o A377506 = generate_sequence(n=25)

%Y Cf. A374856.

%K sign,changed

%O 1,5

%A _Jwalin Bhatt_, Oct 30 2024