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] OEIS: A002207 (denominators) The simplest way to compute Gregory coefficients is to use the recurrence formula with G1 = 1/2.
It implies the finite summation formula where s(n,ℓ) are the signed Stirling numbers of the first kind.
and Schröder's integral formula[19][20] The Gregory coefficients satisfy the bounds given by Johan Steffensen.
Basic series with these numbers include where γ = 0.5772156649... is Euler's constant.
These results are very old, and their history may be traced back to the works of Gregorio Fontana and Lorenzo Mascheroni.
These numbers are strictly alternating Gn(k) = (-1)n-1|Gn(k)| and involved in various expansions for the zeta-functions, Euler's constant and polygamma functions.