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%I #5 Mar 16 2019 20:54:22
%S 1,416,8029248,2188617320448,9219890831036553216,
%T 618951997873353332851408896,664612512289053409746943478501867520,
%U 11417979606286992596912657092388671906224537600,3138550827867416043144948384462236556766662742325141176320,13803492680625189520462719913413857044944571496910203607451430729809920,971334446046166058747728167330455811906524833831361460284791253155406264015674933248
%N G.f. A(x) satisfies: 1 + 4*x = Sum_{n>=0} (4^n + q*A(x))^n * x^n / (1 + 4^n*q*x*A(x))^(n+1), where q = sqrt(128/3).
%H Paul D. Hanna, <a href="/A324613/b324613.txt">Table of n, a(n) for n = 0..50</a>
%F Let q = sqrt(128/3), then g.f. A(x) satisfies:
%F (1) 1 + 4*x = Sum_{n>=0} (4^n + q * A(x))^n * x^n / (1 + 4^n * q * x*A(x))^(n+1).
%F (2) 1 + 4*x = Sum_{n>=0} (4^n - q * A(x))^n * x^n / (1 - 4^n * q * x*A(x))^(n+1).
%e G.f.: A(x) = 1 + 416*x + 8029248*x^2 + 2188617320448*x^3 + 9219890831036553216*x^4 + 618951997873353332851408896*x^5 + 664612512289053409746943478501867520*x^6 + ...
%e Let q = sqrt(128/3), then
%e 1 + 4*x = 1/(1+x*q*A(x)) + (4 + q*A(x))*x/(1 + 4*x*q*A(x))^2 + (4^2 + q*A(x))^2*x^2/(1 + 4^2*x*q*A(x))^3 + (4^3 + q*A(x))^3*x^3/(1 + 4^3*x*q*A(x))^4 + (4^4 + q*A(x))^4*x^4/(1 + 4^4*x*q*A(x))^5 + (4^5 + q*A(x))^5*x^5/(1 + 4^5*x*q*A(x))^6 + (4^6 + q*A(x))^6*x^6/(1 + 4^6*x*q*A(x))^7 + ...
%e and also
%e 1 + 4*x = 1/(1-x*q*A(x)) + (4 - q*A(x))*x/(1 - 4*x*q*A(x))^2 + (4^2 - q*A(x))^2*x^2/(1 - 4^2*x*q*A(x))^3 + (4^3 - q*A(x))^3*x^3/(1 - 4^3*x*q*A(x))^4 + (4^4 - q*A(x))^4*x^4/(1 - 4^4*x*q*A(x))^5 + (4^5 - q*A(x))^5*x^5/(1 - 4^5*x*q*A(x))^6 + (4^6 - q*A(x))^6*x^6/(1 - 4^6*x*q*A(x))^7 + ...
%o (PARI) /* Requires high precision */
%o {a(n) = my(q=sqrt(128/3),A=[1,416,0]); for(i=0,n,
%o A=concat(A,0); A[#A-1] = round( polcoeff( sum(n=0,#A, (4^n + q * Ser(A))^n * x^n / (1 + 4^n * q * x*Ser(A))^(n+1) ),#A)/512));A[n+1]}
%o for(n=0,10, print1(a(n),", "))
%Y Cf. A324299.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 16 2019
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