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A193425 Expansion of e.g.f.: (1 - 2*x)^(-1/(1-x)).

%I A193425 #24 Jul 12 2023 05:52:44
%S A193425 1,2,12,96,976,12000,172608,2838528,52474112,1076451840,24254069760,
%T A193425 595235266560,15801350443008,451082627014656,13778232107286528,
%U A193425 448348123661598720,15483358506138009600,565560454279135887360
%N A193425 Expansion of e.g.f.: (1 - 2*x)^(-1/(1-x)).
%H A193425 G. C. Greubel, <a href="/A193425/b193425.txt">Table of n, a(n) for n = 0..400</a>
%F A193425 E.g.f.: exp( Sum_{n>=1} (2*x)^n/n * Sum_{k=0..n-1} 1/C(n-1,k) ).
%F A193425 E.g.f.: exp( Sum_{n>=1} 2*A126674(n)*x^n/n ), where A126674(n) = n!*Sum_{j=0..n-1} 2^j/(j+1).
%F A193425 a(n) ~ n!*n*2^n * (1 - 2*log(n)/n). - _Vaclav Kotesovec_, Jun 27 2013
%e A193425 E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 96*x^3/3! + 976*x^4/4! + 12000*x^5/5! +...
%e A193425 where the logarithm involves sums of reciprocal binomial coefficients:
%e A193425 log(A(x)) = 2*x*(1) + (2*x)^2/2*(1 + 1) + (2*x)^3/3*(1 + 1/2 + 1) + (2*x)^4/4*(1 + 1/3 + 1/3 + 1) + (2*x)^5/5*(1 + 1/4 + 1/6 + 1/4 + 1) + (2*x)^6/6*(1 + 1/5 + 1/10 + 1/10 + 1/5 + 1) +...
%e A193425 Explicitly, the logarithm begins:
%e A193425 log(A(x)) = 2*x + 8*x^2/2! + 40*x^3/3! + 256*x^4/4! + 2048*x^5/5! + 19968*x^6/6! +...
%e A193425 in which the coefficients equal 2*A126674(n).
%t A193425 CoefficientList[Series[(1-2*x)^(-1/(1-x)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 27 2013 *)
%o A193425 (PARI) {a(n)=n!*polcoeff(exp(sum(m=1,n,2^m*x^m/m*sum(k=0,m-1,1/binomial(m-1,k)))+x*O(x^n)),n)}
%o A193425 (PARI) {a(n)=n!*polcoeff((1-2*x+x*O(x^n))^(-1/(1-x)),n)}
%o A193425 (Magma)
%o A193425 m:=50;
%o A193425 f:= func< x | Exp((&+[(&+[ 1/Binomial(n-1,k): k in [0..n-1]])*(2*x)^n/n: n in [1..m+2]])) >;
%o A193425 R<x>:=PowerSeriesRing(Rationals(), m);
%o A193425 Coefficients(R!(Laplace( f(x) ))); // _G. C. Greubel_, Feb 02 2023
%o A193425 (SageMath)
%o A193425 m=50
%o A193425 def f(x): return exp(sum(sum( 1/binomial(n-1,k) for k in range(n))*(2*x)^n/n for n in range(1,m+2)))
%o A193425 def A193425_list(prec):
%o A193425     P.<x> = PowerSeriesRing(QQ, prec)
%o A193425     return P( f(x) ).egf_to_ogf().list()
%o A193425 A193425_list(m) # _G. C. Greubel_, Feb 02 2023
%Y A193425 Cf. A126674.
%K A193425 nonn
%O A193425 0,2
%A A193425 _Paul D. Hanna_, Jul 27 2011