|
%I #20 May 07 2021 14:09:53
%S 1,6,54,588,7116,92016,1244928,17405520,249486480,3646632288,
%T 54146466528,814458834432,12384344444160,190052162396160,
%U 2939737725858816,45785756862006528,717416350525430016,11301288605493981696,178873923678771712512,2843246259040708414464
%N G.f. satisfies: A(x) = (1 + 2*x*A(x))^2 * (2 + A(x)) / 3.
%H G. C. Greubel, <a href="/A231554/b231554.txt">Table of n, a(n) for n = 0..800</a>
%H Elżbieta Liszewska, Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%F G.f. A(x) satisfies:
%F (1) A(x) = exp( x*(2*A(x) + A(x)^2) + Integral(2*A(x) + A(x)^2 dx) ).
%F (2) A(x) = (1/x)*Series_Reversion( x*(1-2*x-2*x^2)/(1+2*x)^2 ).
%F (3) A(x) = 1 + 2*x*A(x)*(2 + A(x))*(1 + x*A(x)).
%F (4) A(x) = 1 + Sum_{n>=2} (-2)^(n-2) * 3*n * x^(n-1) * A(x)^n.
%F D-finite Recurrence: n*(n+1)*(5*n-7)*a(n) = n*(90*n^2 - 171*n + 67)*a(n-1) - (75*n^3 - 255*n^2 + 254*n - 72)*a(n-2) + 8*(n-3)*(2*n-3)*(5*n-2)*a(n-3). - _Vaclav Kotesovec_, Dec 29 2013
%F a(n) ~ 2^(5*n-4/3) / (n^(3/2) * sqrt(Pi*s) * r^n), where r = 1.862506043468007499... is the root of the equation -2048 + 1152*r - 30*r^2 + r^3 = 0 and s = 0.0684490196162931593... is the root of the equation -125 + 386700*s^3 + 9529446*s^6 + 134217728*s^9 = 0. - _Vaclav Kotesovec_, Dec 29 2013
%e G.f.: A(x) = 1 + 6*x + 54*x^2 + 588*x^3 + 7116*x^4 + 92016*x^5 +...
%e Related expansions.
%e (1 + 2*x*A(x))^2 = 1 + 4*x + 28*x^2 + 264*x^3 + 2928*x^4 + 35760*x^5 +...
%e (2 + A(x))/3 = 1 + 2*x + 18*x^2 + 196*x^3 + 2372*x^4 + 30672*x^5 +...
%e 2*A(x) + A(x)^2 = 3 + 24*x + 252*x^2 + 3000*x^3 + 38436*x^4 + 516960*x^5 +...
%e log(A(x)) = 6*x + 72*x^2/2 + 1008*x^3/3 + 15000*x^4/4 + 230616*x^5/5 +...
%t CoefficientList[1/x*InverseSeries[Series[x*(1-2*x-2*x^2)/(1+2*x)^2, {x, 0, 20}], x],x] (* _Vaclav Kotesovec_, Dec 29 2013 *)
%o (PARI) {a(n)=polcoeff((serreverse(x*(1-2*x-2*x^2)/(1+2*x)^2 +x^2*O(x^n))/x), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1); for(i=1, n, A=exp(x*(2*A+A^2)+intformal(2*A+A^2 +x*O(x^n)))); polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1); for(i=1, n, A=1+2*x*A*(2+A)*(1+x*A) +x*O(x^n)); polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A228966, A231552, A231553, A231556, A231615, A231616, A231618.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 10 2013
| 