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A228714
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G.f. satisfies: A(x) = -1 + x + A(x)^2 + 1/A(x)^2.
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3
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1, 1, 4, 28, 245, 2402, 25235, 277756, 3161523, 36909060, 439516495, 5317812722, 65188448962, 807898166866, 10105819846135, 127423095583448, 1617823588067512, 20665678033803324, 265397856665673136, 3424669486515830710, 44381054246951597644, 577365058173382341896
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OFFSET
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0,3
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COMMENTS
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Compare to the trivial identity:
G(x) = -1 + x + G(x) + 1/G(x) is satisfied when G(x) = 1/(1-x).
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LINKS
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FORMULA
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G.f. satisfies: 0 = 1 - (1-x)*A(x)^2 - A(x)^3 + A(x)^4.
Recurrence: 23*(n-3)*(n-2)*(n-1)*n*(8246*n^2 - 55353*n + 91462)*a(n) = 6*(n-3)*(n-2)*(n-1)*(445284*n^3 - 3656988*n^2 + 9421621*n - 7409572)*a(n-1) - 20*(n-3)*(n-2)*(8246*n^4 - 88337*n^3 + 344800*n^2 - 600301*n + 406224)*a(n-2) - 2*(n-3)*(2*n-7)*(140182*n^4 - 1501729*n^3 + 5829293*n^2 - 9697466*n + 5806800)*a(n-3) + 4*(n-4)*(n-2)*(2*n-9)*(2*n-7)*(8246*n^2 - 38861*n + 44355)*a(n-4). - Vaclav Kotesovec, Sep 08 2013
a(n) ~ sqrt(r*s/(2*Pi*(8*s-3))) / (n^(3/2) * r^n), where s = 1.152776580718308026971178... is the root of the equation s^3*(2*s-1)=2 and r = (2+3*s-4*s^2)/2 = 0.0713771809722745565513... - Vaclav Kotesovec, Mar 31 2014
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EXAMPLE
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G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 245*x^4 + 2402*x^5 + 25235*x^6 +...
where
A(x)^2 = 1 + 2*x + 9*x^2 + 64*x^3 + 562*x^4 + 5518*x^5 + 58018*x^6 +...
1/A(x)^2 = 1 - 2*x - 5*x^2 - 36*x^3 - 317*x^4 - 3116*x^5 - 32783*x^6 -...
so that A(x) = -1 + x + A(x)^2 + 1/A(x)^2.
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MATHEMATICA
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nmax = 22; aa=ConstantArray[0, nmax]; aa[[1]] = 1; Do[AGF = 1 + Sum[aa[[n]] x^n, {n, 1, j-1}] + koef x^j; sol = Solve[Coefficient[1 - (1 - x) AGF^2 - AGF^3 + AGF^4, x, j]==0, koef][[1]]; aa[[j]] = koef/.sol[[1]], {j, 2, nmax}]; Flatten[{1, aa}] (* Vaclav Kotesovec, Sep 08 2013 *)
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PROG
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(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=-1+x + A^2 + 1/(A^2 +x*O(x^n))); polcoeff(A, n)}
for(n=0, 20, 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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EXTENSIONS
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STATUS
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approved
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