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%I #16 Apr 02 2014 17:03:30
%S 1,1,4,28,245,2402,25235,277756,3161523,36909060,439516495,5317812722,
%T 65188448962,807898166866,10105819846135,127423095583448,
%U 1617823588067512,20665678033803324,265397856665673136,3424669486515830710,44381054246951597644,577365058173382341896
%N G.f. satisfies: A(x) = -1 + x + A(x)^2 + 1/A(x)^2.
%C Compare to the trivial identity:
%C G(x) = -1 + x + G(x) + 1/G(x) is satisfied when G(x) = 1/(1-x).
%H Vincenzo Librandi, <a href="/A228714/b228714.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. satisfies: 0 = 1 - (1-x)*A(x)^2 - A(x)^3 + A(x)^4.
%F 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
%F 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
%e G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 245*x^4 + 2402*x^5 + 25235*x^6 +...
%e where
%e A(x)^2 = 1 + 2*x + 9*x^2 + 64*x^3 + 562*x^4 + 5518*x^5 + 58018*x^6 +...
%e 1/A(x)^2 = 1 - 2*x - 5*x^2 - 36*x^3 - 317*x^4 - 3116*x^5 - 32783*x^6 -...
%e so that A(x) = -1 + x + A(x)^2 + 1/A(x)^2.
%t 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 *)
%o (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)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A228713, A228923.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 06 2013
%E Minor edits _Vaclav Kotesovec_, Mar 31 2014
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