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

1, 1, 6, 76, 1620, 51780, 2310000, 136898496, 10393064640, 982930939200, 113269208976000, 15619762139984640, 2539231615282602240, 480507998223110457600, 104704722014993388288000, 26027184253285000629043200, 7320192187611052189440000000, 2312657526289162442074933248000

Offset: 0

Author

Philippe Deléham, Apr 14 1997

Keywords

• nonn

Comments

Related to 'logarithmic numbers'.

Links

• Philippe Deléham, Letter to N. J. A. Sloane, Apr 14 1997
• Wikipedia, Gregory Coefficients

Crossrefs

Cf. A002206, A002207, A061444, A001620.

Programs

Maple

a := n -> local k; (-1)^n*(n+2)!*add(Stirling1(n+1, k)/(k+1), k = 0..n+1):
# Or:
ser := series(1/log(1-x), x, 20): seq((n+1)!*(n+2)!*coeff(ser, x, n), n = 0..17);
# Peter Luschny, Jun 23 2025

Mathematica

Table[(n+2)!*Abs[Sum[StirlingS1[n+1,k]/(k+1),{k,0,n+1}]],{n,0,20}] (* Vaclav Kotesovec, Aug 03 2014 *)
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 *)

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))

Formula

a(n) = (-1)^(n+1)*(n+1)!*(n+2)!*A002206/A002207(n).

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

Sum_{n>=0} a(n)/((n+1)*(n+1)!*(n+2)!) = Euler constant gamma = 0.5772156649... = A001620. - Philippe Deléham, Feb 26 2004

Sum_{n>0} a(n-1)/(n-1)! * x^n/n! = 1 + x/log(1-x). - Michael Somos, Jun 21 2025

Extensions

Better description and more terms from Joe Keane (jgk(AT)jgk.org), Aug 13 2002