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A317339 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - 1/(1+x)^n )^n = 1.

%I A317339 #12 Aug 16 2018 12:24:39
%S A317339 1,1,1,4,26,239,2768,38267,611193,11040954,222241117,4929304517,
%T A317339 119423079917,3137864557135,88884310756274,2700439386780586,
%U A317339 87603920737623984,3022626187893726774,110534722263602544357,4270777627515614565004,173854104446646589718022,7437462737558953036993295
%N A317339 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - 1/(1+x)^n )^n  =  1.
%H A317339 Paul D. Hanna, <a href="/A317339/b317339.txt">Table of n, a(n) for n = 0..200</a>
%F A317339 G.f. A(x) satisfies:
%F A317339 (1) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^n )^n.
%F A317339 (2) A(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(n+1) )^n.
%F A317339 (3) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(n+1) )^n / (1+x)^(n+1).
%F A317339 a(n) ~ n^n / (2^(log(2)/2 + 5/2) * sqrt(1-log(2)) * exp(n) * (log(2))^(2*n + 1)). - _Vaclav Kotesovec_, Aug 12 2018
%e A317339 G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 26*x^4 + 239*x^5 + 2768*x^6 + 38267*x^7 + 611193*x^8 + 11040954*x^9 + 222241117*x^10 + ...
%e A317339 such that
%e A317339 1 = 1  +  (1/A(x) - 1/(1+x))  +  (1/A(x) - 1/(1+x)^2)^2  +  (1/A(x) - 1/(1+x)^3)^3  +  (1/A(x) - 1/(1+x)^4)^4  +  (1/A(x) - 1/(1+x)^5)^5  +  (1/A(x) - 1/(1+x)^6)^6  +  (1/A(x) - 1/(1+x)^7)^7  +  (1/A(x) - 1/(1+x)^8)^8  + ...
%e A317339 Also,
%e A317339 A(x) = 1  +  (1/A(x) - 1/(1+x)^2)  +  (1/A(x) - 1/(1+x)^3)^2  +  (1/A(x) - 1/(1+x)^4)^3  +  (1/A(x) - 1/(1+x)^5)^4  +  (1/A(x) - 1/(1+x)^6)^5  +  (1/A(x) - 1/(1+x)^7)^6  +  (1/A(x) - 1/(1+x)^8)^7  +  (1/A(x) - 1/(1+x)^9)^8  + ...
%o A317339 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( 1/Ser(A) - 1/(1+x)^(m+1) )^m ) )[#A]/2 ); A[n+1]}
%o A317339 for(n=0, 25, print1(a(n), ", "))
%Y A317339 Cf. A317801, A317802, A317803, A317349.
%K A317339 nonn
%O A317339 0,4
%A A317339 _Paul D. Hanna_, Aug 10 2018