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

%I A073479 #34 Sep 08 2022 08:45:06
%S A073479 1,1,4,15,80,490,3534,28938,266048,2710440,30311640,369127440,
%T A073479 4862219592,68881435896,1044331262688,16872336545400,289380447338880,
%U A073479 5251237965683520,100519388543098944,2024241909160239936,42780009017657888640,946724781741392908800
%N A073479 Expansion of e.g.f.: (1-x)^(-1-x).
%H A073479 Robert Israel, <a href="/A073479/b073479.txt">Table of n, a(n) for n = 0..448</a>
%F A073479 (1-x)^(-1-x) = Sum_{n>=0} (Product_{k=1..n} (k+x)) * x^n/n!. [_Paul D. Hanna_, Nov 01 2010]
%F A073479 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
%F A073479 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
%F A073479 a(n) ~ n! * (n - log(n) + 1 - gamma), where gamma is Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Sep 29 2013
%F A073479 E.g.f.: exp( x + Sum_{n>=2} (2*n-1) * x^n / (n*(n-1)) ). - _Paul D. Hanna_, Sep 27 2014
%e A073479 E.g.f.: A(x) = 1 + x + 4*x^2/2! + 15*x^3/3! + 80*x^4/4! + 490*x^5/5! +...
%e A073479 Explicit expressions for the e.g.f.:
%e A073479 (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
%e A073479 (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
%p A073479 S:= series((1-x)^(-1-x),x,51):
%p A073479 seq(coeff(S,x,j)*j!,j=0..50); # _Robert Israel_, Apr 20 2017
%t A073479 CoefficientList[ Series[(1 - x)^(-1 - x), {x, 0, 19}], x]*Table[(n - 1)!, {n, 1, 20}]
%o A073479 (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
%o A073479 (PARI) {a(n)=n!*polcoeff((1-x+x*O(x^n))^(-1-x),n)} \\ _Paul D. Hanna_, Nov 01 2010
%o A073479 (Magma) m:=25; R<x>:=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
%Y A073479 Cf. A007113, A007120.
%K A073479 nonn
%O A073479 0,3
%A A073479 _Vladeta Jovovic_, Aug 26 2002