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%I #20 May 22 2023 02:15:14
%S 1,4,20,106,586,3356,19728,118382,722208,4466050,27931600,176371300,
%T 1122867012,7199842666,46454345844,301384205640,1964899532794,
%U 12866563846920,84585757496444,558060746899684,3693810227983576,24521903234307786,163234951757526400
%N a(n) = coefficient of x^n in A(x) such that 2 = Sum_{n=-oo..+oo} (-x)^n * (2*A(x) + x^(n-1))^(n+1).
%H Paul D. Hanna, <a href="/A359712/b359712.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
%F (1) 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(n-1))^(n+1).
%F (2) 2*x = Sum_{n=-oo..+oo} (-1)^n * (2*x*A(x) + x^n)^(n+1).
%F (3) 2*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^(n-1).
%F (4) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * (2*x*A(x) + x^n)^n ].
%F (5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 2*A(x)*x^(n+1))^n ].
%F From _Paul D. Hanna_, May 12 2023: (Start)
%F (6) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (2*A(x) + x^n)^n.
%F (7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (2*A(x) + x^n)^n ].
%F (8) 2*x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^(n+1).
%F (9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (2*A(x) + x^n)^(n+1).
%F (10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^n)^n.
%F (11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^n. (End)
%F a(n) = Sum_{k=0..n} A359670(n,k)*2^k for n >= 0.
%e G.f.: A(x) = 1 + 4*x + 20*x^2 + 106*x^3 + 586*x^4 + 3356*x^5 + 19728*x^6 + 118382*x^7 + 722208*x^8 + 4466050*x^9 + 27931600*x^10 + ...
%o (PARI) {a(n) = my(A=1,y=2); for(i=1,n,
%o A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
%o polcoeff( A,n,x)}
%o for(n=0,25, print1( a(n),", "))
%o (PARI) {a(n) = my(A=[1],y=2); for(i=1,n, A = concat(A,0);
%o A[#A] = polcoeff(-y + sum(n=-#A,#A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y),#A-1,x) ); A[n+1]}
%o for(n=0,25, print1( a(n),", "))
%Y Cf. A359670, A359711, A359713, A363104, A363105, A361778.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 17 2023
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