A073479 Expansion of e.g.f.: (1-x)^(-1-x).

1, 1, 4, 15, 80, 490, 3534, 28938, 266048, 2710440, 30311640, 369127440, 4862219592, 68881435896, 1044331262688, 16872336545400, 289380447338880, 5251237965683520, 100519388543098944, 2024241909160239936, 42780009017657888640, 946724781741392908800

Offset: 0

Author

Vladeta Jovovic, Aug 26 2002

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 15*x^3/3! + 80*x^4/4! + 490*x^5/5! +...
Explicit expressions for the e.g.f.:
(1-x)^(-1-x) = 1 + (1+x)*x + (1+x)(2+x)*x^2/2! + (1+x)(2+x)(3+x)*x^3/3! +... - _Paul D. Hanna_, Nov 01 2010
(1-x)^(-1-x) = exp(x + 3*x^2/2 + 5*x^3/6 + 7*x^4/12 + 9*x^5/20 + 11*x^6/30 +...). - _Paul D. Hanna_, Sep 27 2014
		

Links

• Robert Israel, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A007113, A007120.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-x)^(-1-x) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 30 2018

Maple

S:= series((1-x)^(-1-x),x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Apr 20 2017

Mathematica

CoefficientList[ Series[(1 - x)^(-1 - x), {x, 0, 19}], x]*Table[(n - 1)!, {n, 1, 20}]

PARI

{a(n)=n!*polcoeff(sum(m=0,n,prod(k=1,m,k+x)*x^m/m!)+x*O(x^n),n)} \\ Paul D. Hanna, Nov 01 2010

PARI

{a(n)=n!*polcoeff((1-x+x*O(x^n))^(-1-x),n)} \\ Paul D. Hanna, Nov 01 2010

Formula

(1-x)^(-1-x) = Sum_{n>=0} (Product_{k=1..n} (k+x)) * x^n/n!. [Paul D. Hanna, Nov 01 2010]

E.g.f.: (1-x)^(-1-x) = 1+(x*(1+x))/(Q(0)-x*(1+x)); Q(k)=(1+x)*k+1+x+(x^2)-x*(k+1)*(k+2+x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2011

E.g.f.: 1 + x*(Q(0) - 1)/(x-1) where Q(k) = 1 - (1+x/(k+1))/(1 - x/(x - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 05 2013

a(n) ~ n! * (n - log(n) + 1 - gamma), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 29 2013

E.g.f.: exp( x + Sum_{n>=2} (2*n-1) * x^n / (n*(n-1)) ). - Paul D. Hanna, Sep 27 2014