Gregory coefficients
Gregory coefficients Gn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the rational numbers that occur in the Maclaurin series expansion of the reciprocal logarithm z ln ( 1 + z ) = 1 + 1 2 z − 1 12 z 2 + 1 24 z 3 − 19 720 z 4 + 3 160 z 5 − 863 60480 z 6 + ⋯ = 1 + ∑ n = 1 ∞ G n z n , | z | < 1 . {\displaystyle {\begin{aligned}{\frac {z}{\ln(1+z)}}&=1+{\frac {1}{2}}z-{\frac {1}{12}}z^{2}+{\frac {1}{24}}z^{3}-{\frac {19}{720}}z^{4}+{\frac {3}{160}}z^{5}-{\frac {863}{60480}}z^{6}+\cdots \\&=1+\sum _{n=1}^{\infty }G_{n}z^{n}\,,\qquad |z|<1\,.\end{aligned}}} Gregory coefficients are alternating Gn = (−1)n−1|Gn| for n > 0 and decreasing in absolute value.