A250918 - OEIS

%I #23 Jun 02 2017 21:24:12

%S 1,1,2,6,19,67,244,928,3608,14348,57936,237228,982006,4104266,

%T 17292220,73373704,313255939,1344703363,5800347378,25128400374,

%U 109287352255,476990070863,2088544907796,9171761291584,40385942346364,178272020615752,788731516970684,3496994698151688,15535135825359760,69140216020170812

%N G.f. A(x) satisfies: 2*A(x) = 1 + A(x)^3/(1 + x*A(x)^3).

%H G. C. Greubel, <a href="/A250918/b250918.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. A(x) satisfies:

%F (1) 1 - 2*A(x) + (1+x)*A(x)^3 - 2*x*A(x)^4 = 0.

%F (2) (1 - 2*A(x) + A(x)^3) / (2*A(x)^4 - A(x)^3) = x.

%F (3) 1/(1 + x*A(x)^3) + 2*x*A(x) = 1 + x.

%F (4) A(x) = 1 + Series_Reversion( x*(1 + 3*x + x^2) / ((1+x)^3 * (1+2*x)) ).

%F a(n) ~ sqrt(8 - 5*sqrt(2/3) + 3*sqrt(9-3*sqrt(6))) / (4 * n^(3/2) * sqrt(Pi) * (2*sqrt(6) - 5 + 4/sqrt(81 + 33*sqrt(6)))^n). - _Vaclav Kotesovec_, Jan 17 2015

%e G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 67*x^5 + 244*x^6 + 928*x^7 +...

%e where

%e 2*A(x) = 1 + A(x)^3 - x*A(x)^6 + x^2*A(x)^9 - x^3*A(x)^12 + x^4*A(x)^15 - x^5*A(x)^18 + x^6*A(x)^21 - x^7*A(x)^24 +-...

%e Also

%e 2*A(x) = 1 + (1+x)*A(x)^3 - 2*x*A(x)^4.

%t CoefficientList[1 + InverseSeries[Series[x*(1 + 3*x + x^2) / ((1+x)^3 * (1+2*x)), {x, 0, 20}], x],x] (* _Vaclav Kotesovec_, Jan 17 2015 *)

%o (PARI) /* Generates vector A of initial terms: */

%o {A=[1,1];for(i=1,40,A=concat(A,0);A[#A]=Vec(sum(n=1,#A,(-x)^n*Ser(A)^(3*n)))[#A]);A}

%o (PARI) /* G.f.: 1 + Series_Reversion(x*(1+3*x+x^2)/((1+x)^3*(1+2*x))) */

%o {a(n)=local(A= 1 + serreverse(x*(1 + 3*x + x^2)/((1+x)^3*(1+2*x)) +x^2*O(x^n))); polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 15 2015