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%I #69 Sep 08 2022 08:45:17
%S 0,1,1,2,6,24,120,720,5040,40320,362880,3628800,39916800,479001600,
%T 6227020800,87178291200,1307674368000,20922789888000,355687428096000,
%U 6402373705728000,121645100408832000,2432902008176640000,51090942171709440000,1124000727777607680000
%N Shifted factorial numbers: a(0)=0, a(n) = (n-1)!.
%C E.g.f.: Sum_{n>=1} (n-1)!*x^n/n! = Sum_{n>=1} x^n/n.
%C The shift law of the e.g.f.: if Sum_{n>=0} a(n)*x^n/n! = f(x), then Sum_{n>=0} a(n+1)*x^n/n! = d/dx f(x) and Sum_{n>=1} a(n-1)*x^n/n! = Integral f(x) dx.
%C The e.g.f. of A000142 (= n!) is 1/(1-x), so the e.g.f. of a(n)=(n-1)! is integral 1/(1-x) = -log(1-x).
%D A. N. Khovanskii. The Application of Continued Fractions and Their Generalizations to Problem in Approximation Theory. Groningen: Noordhoff, Netherlands, 1963. See p.141 (10.19)
%H Vincenzo Librandi, <a href="/A104150/b104150.txt">Table of n, a(n) for n = 0..200</a>
%H S. N. Gladkovskii, <a href="http://www.archive.org/details/AnalysisOfTheContinuedFractions.2.3-ed.2009-GladkovskiiSergeiNikolaevich">Analysis Of The Continued Fractions</a> (in Russian), see p.79 (5.1.21)
%F E.g.f. -log(1-x) = x + x^2/2 + x^3/3 + ... + x^n/n + ...
%F G.f.: x+x^2/(G(0)-x) where G(k) = 1 - (k+1)*x/(1 - x*(k+2)/G(k+1)); G(0) = W(1,1;-x)/W(1,2;-x), W(a,b,x) = 1 - a*b*x/1! + a*(a+1)*b*(b+1)*x^2/2! - ... + a*(a+1)*...*(a+n-1)*b*(b+1)*...*(b+n-1)*x^n/n! + ...; see [A. N. Khovanskii, p. 141 (10.19)], x-> -x; (continued fraction, 2-step). - _Sergei N. Gladkovskii_, Aug 14 2012
%F E.g.f.: (-x + 5*x^2/2 - 11*x^3/6 + x^4/4 + x^5/(W(0)-x)/4)/(x-1)^3 where W(k)= (x + 1)*k + x + 5 - x*(k+2)*(k+5)/W(k+1); see [S. N. Gladkovskii, p. 79 (5.1.21)]; (continued fraction, Euler's 1st kind, 1-step). - _Sergei N. Gladkovskii_, Aug 15 2012
%F G.f.: A(x) = Integral_{t>=0} x*exp(-t)/(1-x*t) dt = x/G(0) where G(k) = 1 - x*(k+1)/(1 - x*(k+1)/G(k+1) )); (continued fraction due to L. Euler and E. N. Laguerre). - _Sergei N. Gladkovskii_, Dec 24 2012
%F G.f.: x + x/Q(0), where Q(k)= 1/x - (2*k+2) - (k+2)*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Apr 25 2013
%F G.f.: x/Q(0), where Q(k) = 1 - x*(k+1)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, May 20 2013
%F G.f.: x*G(0), where G(k) = 1 + x*(2*k+1)/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 07 2013
%F G.f.: x*G(0), where G(k) = 1 - x*(k+1)/(x*(k+1) - 1/(1 - x*(k+1)/(x*(k+1) - 1/G(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Aug 07 2013
%t Join[{0,1},Range[20]!] (* _Harvey P. Dale_, Dec 09 2013 *)
%o (Sage) [stirling_number1(n,1) for n in range(0, 22)] # _Zerinvary Lajos_, May 16 2009
%o (Magma) [0] cat [Factorial(n-1): n in [1..25]]; // _Vincenzo Librandi_, Dec 25 2012
%o (PARI) x='x+O('x^30); concat([0], Vec(serlaplace(-log(1-x)))) \\ _G. C. Greubel_, May 15 2018
%Y Cf. A000142.
%Y Column k=1 of A285849.
%Y Main diagonal of A295027 (for n > 0).
%K easy,nonn
%O 0,4
%A _Miklos Kristof_, Mar 08 2005
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