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A203852 Expansion of e.g.f. exp( Integral -log(1-x) dx ).

%I A203852 #29 Sep 08 2022 08:46:01
%S A203852 1,0,1,1,5,16,79,421,2673,19216,156021,1411873,14117773,154730720,
%T A203852 1845959179,23826445501,330951133537,4923574598112,78123812086441,
%U A203852 1317174439409409,23517962293307701,443340968936640496,8799729204814165223,183448995762912568885
%N A203852 Expansion of e.g.f. exp( Integral -log(1-x) dx ).
%C A203852 Row sums of A185164. - _Peter Bala_, Mar 14 2012
%H A203852 G. C. Greubel, <a href="/A203852/b203852.txt">Table of n, a(n) for n = 0..450</a>
%F A203852 E.g.f.: exp( Sum_{n>=2} x^n/(n*(n-1)) ).
%F A203852 E.g.f.: exp(x)*(1-x)^(1-x). - _Vaclav Kotesovec_, Dec 27 2013
%F A203852 a(n) ~ (n-1)! * (exp(1)/n + (2*log(n)+2*gamma)/n^2), where gamma is the Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Dec 27 2013
%F A203852 a(n) = sum(i=1..n-1, binomial(n-1,i)*(i-1)!*a(n-i-1),i,1,n-1), a(0)=1. - _Vladimir Kruchinin_, Feb 23 2015
%e A203852 E.g.f.: A(x) = 1 + x^2/2! + x^3/3! + 5*x^4/4! + 16*x^5/5! + 79*x^6/6! +...
%e A203852 where: log(A(x)) = x^2/2 + x^3/6 + x^4/12 + x^5/20 + x^6/30 + x^7/42 +...
%t A203852 CoefficientList[Series[Exp[x]*(1-x)^(1-x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Dec 27 2013 *)
%o A203852 (PARI) {a(n)=n!*polcoeff(exp(-intformal(log(1-x +x*O(x^n)))), n)}
%o A203852 (PARI) x='x+O('x^30); Vec(serlaplace(exp(x)*(1-x)^(1-x))) \\ _G. C. Greubel_, Jul 17 2018
%o A203852 (Maxima)
%o A203852 a(n):=if n=0 then 1 else sum(binomial(n-1,i)*(i-1)!*a(n-i-1),i,1,n-1); /* _Vladimir Kruchinin_, Feb 23 2015 */
%o A203852 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)*(1-x)^(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Jul 17 2018
%Y A203852 Cf. A185164.
%K A203852 nonn
%O A203852 0,5
%A A203852 _Paul D. Hanna_, Jan 29 2012