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A324613
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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).
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3
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1, 416, 8029248, 2188617320448, 9219890831036553216, 618951997873353332851408896, 664612512289053409746943478501867520, 11417979606286992596912657092388671906224537600, 3138550827867416043144948384462236556766662742325141176320, 13803492680625189520462719913413857044944571496910203607451430729809920, 971334446046166058747728167330455811906524833831361460284791253155406264015674933248
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OFFSET
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0,2
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LINKS
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FORMULA
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Let q = sqrt(128/3), then g.f. A(x) satisfies:
(1) 1 + 4*x = Sum_{n>=0} (4^n + q * A(x))^n * x^n / (1 + 4^n * q * x*A(x))^(n+1).
(2) 1 + 4*x = Sum_{n>=0} (4^n - q * A(x))^n * x^n / (1 - 4^n * q * x*A(x))^(n+1).
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EXAMPLE
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G.f.: A(x) = 1 + 416*x + 8029248*x^2 + 2188617320448*x^3 + 9219890831036553216*x^4 + 618951997873353332851408896*x^5 + 664612512289053409746943478501867520*x^6 + ...
Let q = sqrt(128/3), then
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 + ...
and also
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 + ...
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PROG
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(PARI) /* Requires high precision */
{a(n) = my(q=sqrt(128/3), A=[1, 416, 0]); for(i=0, n,
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]}
for(n=0, 10, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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