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A317349 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^n )^n = 1.

%I A317349 #31 Aug 12 2018 05:09:45
%S A317349 1,1,2,7,42,372,4269,59047,946557,17175289,347208299,7730688884,
%T A317349 187911183701,4951155672353,140575561645293,4279249948000903,
%U A317349 139050095246322895,4804391579357016747,175902340755219278039,6803436418471129704925,277202774381386656583959,11868116969794805874111831
%N A317349 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^n )^n  =  1.
%H A317349 Paul D. Hanna, <a href="/A317349/b317349.txt">Table of n, a(n) for n = 0..300</a>
%F A317349 G.f. A(x) satisfies:
%F A317349 (1) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^n )^n.
%F A317349 (2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(n+1) )^n.
%F A317349 (3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(n+1) )^n * (1-x)^(n+1).
%F A317349 (4) A(x)^2 = 2*A(x) * [ Sum_{n>=0} (n+1) * ( 1/A(x) - (1-x)^(n+1) )^n ] - [ Sum_{n>=0} (n+1) * ( 1/A(x) - (1-x)^(n+2) )^n ].
%F A317349 (5) A(x) = [ Sum_{n>=1} n*(n+1)/2 * (1-x)^(n+1) * ( 1/Ser(A) - (1-x)^(n+1) )^(n-1) ] / [ Sum_{n>=1} n^2 * (1-x)^n * ( 1/Ser(A) - (1-x)^n )^(n-1) ].
%F A317349 a(n) ~ 2^(log(2)/2 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - _Vaclav Kotesovec_, Aug 06 2018
%e A317349 G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 42*x^4 + 372*x^5 + 4269*x^6 + 59047*x^7 + 946557*x^8 + 17175289*x^9 + 347208299*x^10 + ...
%e A317349 such that
%e A317349 1 = 1  +  (1/A(x) - (1-x))  +  (1/A(x) - (1-x)^2)^2  +  (1/A(x) - (1-x)^3)^3  +  (1/A(x) - (1-x)^4)^4  +  (1/A(x) - (1-x)^5)^5  +  (1/A(x) - (1-x)^6)^6  +  (1/A(x) - (1-x)^7)^7  +  (1/A(x) - (1-x)^8)^8  + ...
%e A317349 Also,
%e A317349 A(x) = 1  +  (1/A(x) - (1-x)^2)  +  (1/A(x) - (1-x)^3)^2  +  (1/A(x) - (1-x)^4)^3  +  (1/A(x) - (1-x)^5)^4  +  (1/A(x) - (1-x)^6)^5  +  (1/A(x) - (1-x)^7)^6  +  (1/A(x) - (1-x)^8)^7  +  (1/A(x) - (1-x)^9)^8  + ...
%o A317349 (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-x)^(m+1) )^m ) )[#A]/2 ); A[n+1]}
%o A317349 for(n=0,25, print1(a(n),", "))
%Y A317349 Cf. A317339, A317348, A317340, A317666, A317667 A317668.
%K A317349 nonn
%O A317349 0,3
%A A317349 _Paul D. Hanna_, Aug 02 2018