A104150 Shifted factorial numbers: a(0)=0, a(n) = (n-1)!.

0, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset

0,4

Comments

E.g.f.: Sum_{n>=1} (n-1)!*x^n/n! = Sum_{n>=1} x^n/n.

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.

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).

References

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)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

S. N. Gladkovskii, Analysis Of The Continued Fractions (in Russian), see p.79 (5.1.21)

Formula

E.g.f. -log(1-x) = x + x^2/2 + x^3/3 + ... + x^n/n + ...

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

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

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

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

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

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

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

Mathematica

Join[{0, 1}, Range[20]!] (* Harvey P. Dale, Dec 09 2013 *)

Prog

(Sage) [stirling_number1(n, 1) for n in range(0, 22)] # Zerinvary Lajos, May 16 2009
(Magma) [0] cat [Factorial(n-1): n in [1..25]]; // Vincenzo Librandi, Dec 25 2012
(PARI) my(x='x+O('x^30)); concat([0], Vec(serlaplace(-log(1-x)))) \\ G. C. Greubel, May 15 2018

Crossrefs

Cf. A000142.

Column k=1 of A285849.

Main diagonal of A295027 (for n > 0).

Sequence in context: A159333 A165233 A000142 * A358185 A370360 A074166

Adjacent sequences: A104147 A104148 A104149 * A104151 A104152 A104153

Keyword

easy,nonn,changed

Author

Miklos Kristof, Mar 08 2005

Status

approved