Gregory coefficients

Gregory coefficients G_n, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind,^{[1][2][3][4][5][6][7][8][9][10][11][12][13]} are the rational numbers that occur in the Maclaurin series expansion of the reciprocal logarithm

${\displaystyle {\begin{aligned}{\frac {z}{\ln(1+z)}}&=1+{\frac {1}{2}}z-{\frac {1}{12}}z^{2}+{\frac {1}{24}}z^{3}-{\frac {19}{720}}z^{4}+{\frac {3}{160}}z^{5}-{\frac {863}{60480}}z^{6}+\cdots \\&=1+\sum _{n=1}^{\infty }G_{n}z^{n}\,,\qquad |z|<1\,.\end{aligned}}}$

Gregory coefficients are alternating G_n = (−1)ⁿ⁻¹|G_n| for n > 0 and decreasing in absolute value. These numbers are named after James Gregory who introduced them in 1670 in the numerical integration context. They were subsequently rediscovered by many mathematicians and often appear in works of modern authors, who do not always recognize them.^{[1][5][14][15][16][17]}