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%I #11 Sep 01 2014 17:23:53
%S 1,4,18,96,624,4896,45072,474624,5619456,73903104,1069106688,
%T 16872800256,288483876864,5311773904896,104789944829952,
%U 2205099306123264,49302137885884416,1167150882577711104,29165495002777387008,767163772371852066816,21188300138891474632704
%N E.g.f. satisfies: A'(x) = 1 + 4*A(x) + A(x)^2, where A(0)=0.
%F E.g.f.: Series_Reversion( Integral 1/(1 + 4*x + x^2) dx ).
%F E.g.f.: (2-sqrt(3))*(exp(2*sqrt(3)*x)-1)/(1 + (4*sqrt(3) - 7)*exp(2*sqrt(3)*x)). - _Vaclav Kotesovec_, Jan 13 2014
%F a(n) ~ n! * ((2*sqrt(3))/log(7+4*sqrt(3)))^(n+1). - _Vaclav Kotesovec_, Jan 13 2014
%F O.g.f.: x/(1-4*x - 1*2*x^2/(1-8*x - 2*3*x^2/(1-12*x - 3*4*x^2/(1-16*x - 4*5*x^2/(1-20*x - 5*6*x^2/(1- .../(1-4*n*x - n*(n+1)*x^2/(1- ...))))))) (continued fraction). - _Paul D. Hanna_, Sep 01 2014
%e E.g.f.: A(x) = x + 4*x^2/2! + 18*x^3/3! + 96*x^4/4! + 624*x^5/5! +...
%e Related series.
%e A(x)^2 = 2*x^2/2! + 24*x^3/3! + 240*x^4/4! + 2400*x^5/5! + 25488*x^6/6! +...
%t Rest[FullSimplify[CoefficientList[Series[(2-Sqrt[3])*(E^(2*Sqrt[3]*x)-1)/(1 + (4*Sqrt[3]-7)*E^(2*Sqrt[3]*x)),{x,0,15}],x]*Range[0,15]!]] (* _Vaclav Kotesovec_, Jan 13 2014 *)
%o (PARI) {a(n)=local(A=x);for(i=1,n,A=intformal(1+4*A+A^2 +x*O(x^n))); n!*polcoeff(A,n)}
%o for(n=1,25,print1(a(n),", "))
%o (PARI) {a(n)=local(A=serreverse(intformal(1/(1+4*x+x^2 +x*O(x^n))))); n!*polcoeff(A,n)}
%o for(n=1,25,print1(a(n),", "))
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jan 09 2014
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