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%I #4 Feb 06 2019 22:49:42
%S 1,2,20,320,7996,269272,11293792,563253696,32433009160,2113347523336,
%T 153579286783456,12309659862402976,1078628781953636960,
%U 102578628758305245024,10523148808846566898816,1158407291029244188955264,136214299772837816557703120,17040721610970237566148646464,2260018461602151565512432884608,316748455363386162460484685488512
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} 2^n * ((1+x)^n - 1)^n / (A(x) + 2 - 2*(1+x)^n)^(n+1).
%H Paul D. Hanna, <a href="/A323574/b323574.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} 2^n * ((1+x)^n - 1)^n/(A(x) + 2 - 2*(1+x)^n)^(n+1).
%F (2) 1 = Sum_{n>=0} 2^n * ((1+x)^n + 1)^n/(A(x) + 2 + 2*(1+x)^n)^(n+1).
%e G.f.: A(x) = 1 + 2*x + 20*x^2 + 320*x^3 + 7996*x^4 + 269272*x^5 + 11293792*x^6 + 563253696*x^7 + 32433009160*x^8 + 2113347523336*x^9 + 153579286783456*x^10 + ...
%e such that
%e 1 = 1/A(x) + 2*((1+x) - 1)/(A(x) + 2 - 2*(1+x))^2 + 2^2*((1+x)^2 - 1)^2/(A(x) + 2 - 2*(1+x)^2)^3 + 2^3*((1+x)^3 - 1)^3/(A(x) + 2 - 2*(1+x)^3)^4 + 2^4*((1+x)^4 - 1)^4/(A(x) + 2 - 2*(1+x)^4)^5 + 2^5*((1+x)^5 - 1)^5/(A(x) + 2 - 2*(1+x)^5)^6 + ...
%e also,
%e 1 = 1/(A(x) + 4) + 2*(1 + (1+x))/(A(x) + 2 + 2*(1+x))^2 + 2^2*(1 + (1+x)^2)^2/(A(x) + 2 + 2*(1+x)^2)^3 + 2^3*(1 + (1+x)^3)^3/(A(x) + 2 + 2*(1+x)^3)^4 + 2^4*(1 + (1+x)^4)^4/(A(x) + 2 + 2*(1+x)^4)^5 + 2^5*(1 + (1+x)^5)^5/(A(x) + 2 + 2*(1+x)^5)^6 + ...
%o (PARI) {a(n) = my(A=[1],X=x+x*O(x^n)); for(i=1,n, A=concat(A,0); A[#A] = Vec( sum(m=0, #A, 2^m * ((1+X)^m - 1)^m / (Ser(A) + 2 - 2*(1+X)^m)^(m+1) ) )[#A]); A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A323313.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 06 2019
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