cp's OEIS Frontend

A009763 a(n) is (n+1)!*(n+2)! times coefficient of x^n in (log(1-x))^-1.

%I A009763 #29 Jun 23 2025 05:33:48
%S A009763 1,1,6,76,1620,51780,2310000,136898496,10393064640,982930939200,
%T A009763 113269208976000,15619762139984640,2539231615282602240,
%U A009763 480507998223110457600,104704722014993388288000,26027184253285000629043200,7320192187611052189440000000,2312657526289162442074933248000
%N A009763 a(n) is (n+1)!*(n+2)! times coefficient of x^n in (log(1-x))^-1.
%C A009763 Related to 'logarithmic numbers'.
%H A009763 Philippe Deléham, <a href="/A009763/a009763.pdf">Letter to N. J. A. Sloane, Apr 14 1997</a>
%H A009763 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gregory_coefficients">Gregory Coefficients</a>
%F A009763 a(n) = (-1)^(n+1)*(n+1)!*(n+2)!*A002206/A002207(n).
%F A009763 log(2*Pi) = 1 + Sum_{n>0}{a(n)*(2n+1)/(((n+1)!)^2*n*(n+1)) = 1.83787706... = A061444. - _Philippe Deléham_, Jan 20 2004
%F A009763 Sum_{n>=0} a(n)/((n+1)*(n+1)!*(n+2)!) = Euler constant gamma = 0.5772156649... = A001620. - _Philippe Deléham_, Feb 26 2004
%F A009763 Sum_{n>0} a(n-1)/(n-1)! * x^n/n! = 1 + x/log(1-x). - _Michael Somos_, Jun 21 2025
%p A009763 a := n -> local k; (-1)^n*(n+2)!*add(Stirling1(n+1, k)/(k+1), k = 0..n+1):
%p A009763 # Or:
%p A009763 ser := series(1/log(1-x), x, 20): seq((n+1)!*(n+2)!*coeff(ser, x, n), n = 0..17);
%p A009763 # _Peter Luschny_, Jun 23 2025
%t A009763 Table[(n+2)!*Abs[Sum[StirlingS1[n+1,k]/(k+1),{k,0,n+1}]],{n,0,20}] (* _Vaclav Kotesovec_, Aug 03 2014 *)
%t A009763 a[ n_] := If[n<0, 0, (n+2)!*(n+1)!*SeriesCoefficient[ 1/x + 1/Log[1-x], {x, 0, n}]]; (* _Michael Somos_, Jun 21 2025 *)
%o A009763 (PARI) a(n)=local(A); if(n<0,0,n++; A=x/log(1-x+x^2*O(x^n)); n!*(n+1)!*polcoeff(A,n))
%Y A009763 Cf. A002206, A002207, A061444, A001620.
%K A009763 nonn
%O A009763 0,3
%A A009763 _Philippe Deléham_, Apr 14 1997
%E A009763 Better description and more terms from Joe Keane (jgk(AT)jgk.org), Aug 13 2002