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A324620
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G.f. A(x) = Sum_{n>=0} x^n*(A(x)^n + sqrt(2))^n/(1 + sqrt(2)*x*A(x)^n)^(n+1).
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0
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1, 1, 2, 3, 7, 22, 92, 440, 2281, 12493, 71765, 430651, 2693870, 17539117, 118715136, 834501419, 6086215582, 46008381159, 360118132277, 2915376645033, 24382943109124, 210428145134993, 1871657093564422, 17137046392867008, 161334811357004021, 1559974013680482615, 15475431343550867360, 157351220236056642597, 1638298847460905532116, 17451514257450951873260, 190036890804241449007404, 2113879533614295483104495
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n*(A(x)^n + sqrt(2))^n / (1 + sqrt(2)*x*A(x)^n)^(n+1).
(2) A(x) = Sum_{n>=0} x^n*(A(x)^n - sqrt(2))^n / (1 - sqrt(2)*x*A(x)^n)^(n+1).
(3) A(x) = Sum_{n>=0} x^n*Sum_{k=0..n} sqrt(2)^k * binomial(n,k) * (A(x)^n - sqrt(2)*A(x)^k)^(n-k).
(4) A(x) = Sum_{n>=0} x^n*Sum_{k=0..n} (-sqrt(2))^k * binomial(n,k) * (A(x)^n + sqrt(2)*A(x)^k)^(n-k). (End)
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 22*x^5 + 92*x^6 + 440*x^7 + 2281*x^8 + 12493*x^9 + 71765*x^10 + 430651*x^11 + 2693870*x^12 + ...
Let q = sqrt(2), then
A(x) = 1/(1 + q*x) + x*(A(x) + q)/(1 + q*x*A(x))^2 + x^2*(A(x)^2 + q)^2 / (1 + q*x*A(x)^2)^3 + x^3*(A(x)^3 + q)^3/(1 + q*x*A(x)^3)^4 + x^4*(A(x)^4 + q)^4 / (1 + q*x*A(x)^4)^5 + x^5*(A(x)^5 + q)^5/(1 + q*x*A(x)^5)^6 + ...
also
A(x) = 1/(1 - q*x) + x*(A(x) - q)/(1 - q*x*A(x))^2 + x^2*(A(x)^2 - q)^2 / (1 - q*x*A(x)^2)^3 + x^3*(A(x)^3 - q)^3/(1 - q*x*A(x)^3)^4 + x^4*(A(x)^4 - q)^4 / (1 - q*x*A(x)^4)^5 + x^5*(A(x)^5 - q)^5/(1 - q*x*A(x)^5)^6 + ...
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PROG
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(PARI) {a(n) = my(A=[1, 1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff( sum(n=0, #A+1, x^n*(Ser(A)^n + sqrt(2))^n/(1 + sqrt(2)*x*Ser(A)^n)^(n+1) ), #A-1)); round( polcoeff(Ser(A), n) )}
for(n=0, 40, 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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