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A317666 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n) )^n = 1.

%I A317666 #13 Aug 13 2018 03:53:47
%S A317666 1,2,7,48,590,10602,244457,6767792,216875258,7863473864,317632851912,
%T A317666 14132208327052,686514289288897,36154193924315170,2051928741855927465,
%U A317666 124870207134047889232,8112089716821244526285,560396754826502247713090,41024663835523296400398275,3172738829903313189522259140,258493327059457440608140711531
%N A317666 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n) )^n  =  1.
%H A317666 Paul D. Hanna, <a href="/A317666/b317666.txt">Table of n, a(n) for n = 0..200</a>
%F A317666 G.f. A(x) satisfies:
%F A317666 (1) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n) )^n.
%F A317666 (2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+2) )^n.
%F A317666 (3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+2) )^n * (1-x)^(2*n+2).
%F A317666 (4) Let B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+1) )^n ,
%F A317666 then B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+2) )^n * (1-x)^(n+1).
%F A317666 a(n) ~ 2^(n + log(2)/4 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - _Vaclav Kotesovec_, Aug 13 2018
%e A317666 G.f.: A(x) = 1 + 2*x + 7*x^2 + 48*x^3 + 590*x^4 + 10602*x^5 + 244457*x^6 + 6767792*x^7 + 216875258*x^8 + 7863473864*x^9 + 317632851912*x^10 + ...
%e A317666 such that
%e A317666 1 = 1  +  (1/A(x) - (1-x)^2)  +  (1/A(x) - (1-x)^4)^2  +  (1/A(x) - (1-x)^6)^3  +  (1/A(x) - (1-x)^8)^4  +  (1/A(x) - (1-x)^10)^5  +  (1/A(x) - (1-x)^12)^6  +  (1/A(x) - (1-x)^14)^7  +  (1/A(x) - (1-x)^16)^8  + ...
%e A317666 Also,
%e A317666 A(x) = 1  +  (1/A(x) - (1-x)^4)  +  (1/A(x) - (1-x)^6)^2  +  (1/A(x) - (1-x)^8)^3  +  (1/A(x) - (1-x)^10)^4  +  (1/A(x) - (1-x)^12)^5  +  (1/A(x) - (1-x)^14)^6  +  (1/A(x) - (1-x)^16)^7  +  (1/A(x) - (1-x)^18)^8  + ...
%e A317666 RELATED SERIES.
%e A317666 The related series B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+1) )^n begins
%e A317666 B(x) = 1 + x + 3*x^2 + 20*x^3 + 245*x^4 + 4394*x^5 + 101203*x^6 + 2800620*x^7 + 89739208*x^8 + 3253949840*x^9 + 131451064170*x^10 + ...
%e A317666 restated,
%e A317666 B(x) = 1  +  (1/A(x) - (1-x)^3)  +  (1/A(x) - (1-x)^5)^2  +  (1/A(x) - (1-x)^7)^3  +  (1/A(x) - (1-x)^9)^4  +  (1/A(x) - (1-x)^11)^5  +  (1/A(x) - (1-x)^13)^6  +  (1/A(x) - (1-x)^15)^7  +  (1/A(x) - (1-x)^17)^8  + ...
%e A317666 which also equals
%e A317666 B(x) = (1-x)  +  (1/A(x) - (1-x)^4)*(1-x)^2  +  (1/A(x) - (1-x)^6)^2*(1-x)^3  +  (1/A(x) - (1-x)^8)^3*(1-x)^4  +  (1/A(x) - (1-x)^10)^4*(1-x)^5  +  (1/A(x) - (1-x)^12)^5*(1-x)^6  +  (1/A(x) - (1-x)^14)^6*(1-x)^7  +  (1/A(x) - (1-x)^16)^7*(1-x)^8  +  (1/A(x) - (1-x)^18)^8*(1-x)^9  + ...
%e A317666 Compare the above to
%e A317666 1 = (1-x)^2  +  (1/A(x) - (1-x)^4)*(1-x)^4  +  (1/A(x) - (1-x)^6)^2*(1-x)^6  +  (1/A(x) - (1-x)^8)^3*(1-x)^8  +  (1/A(x) - (1-x)^10)^4*(1-x)^10  +  (1/A(x) - (1-x)^12)^5*(1-x)^12  +  (1/A(x) - (1-x)^14)^6*(1-x)^14  +  (1/A(x) - (1-x)^16)^7*(1-x)^16  +  (1/A(x) - (1-x)^18)^8*(1-x)^18  + ...
%o A317666 (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)^(2*m+2) )^m ) )[#A]/2 ); A[n+1]}
%o A317666 for(n=0, 25, print1(a(n), ", "))
%Y A317666 Cf. A317349, A317667, A317668, A317801.
%K A317666 nonn
%O A317666 0,2
%A A317666 _Paul D. Hanna_, Aug 12 2018