Gregory coefficients G_n, 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

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.

The simplest way to compute Gregory coefficients is to use the recurrence formula

with G₁ = ⁠1/2⁠. Gregory coefficients may be also computed explicitly via the following differential

or the integral

which can be proved by integrating ${\displaystyle (1+z)^{x}}$ between 0 and 1 with respect to ${\displaystyle x}$, once directly and the second time using the binomial series expansion first.

It implies the finite summation formula

where s(n,ℓ) are the signed Stirling numbers of the first kind.