A218168 - OEIS

%I #8 Aug 24 2017 05:48:05

%S 1,1,2,5,16,77,630,7956,132480,2664389,62140078,1643056625,

%T 48503833280,1580192424580,56292381309608,2176251789724500,

%U 90726545080501440,4056755120695005717,193649641793538259950,9828338731644277922679,528459404673048692682384

%N G.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^3]/A(x)^4.

%H Vaclav Kotesovec, <a href="/A218168/b218168.txt">Table of n, a(n) for n = 0..380</a>

%F G.f. satisfies: A(x)^3 = A(x)^2 + x*A(x) + 3*x^2*A'(x).

%F a(n) ~ c * 3^n * n! / n^2, where c = 0.0220701841954367321830819129913... - _Vaclav Kotesovec_, Aug 24 2017

%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 77*x^5 + 630*x^6 + 7956*x^7 +...

%e Related expansions:

%e A'(x) = 1 + 4*x + 15*x^2 + 64*x^3 + 385*x^4 + 3780*x^5 + 55692*x^6 +...

%e A(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 46*x^4 + 206*x^5 + 1503*x^6 +...

%e A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 96*x^4 + 414*x^5 + 2735*x^6 +...

%e A(x)^4 = 1 + 4*x + 14*x^2 + 48*x^3 + 173*x^4 + 736*x^5 + 4486*x^6 +...

%e d/dx x*A(x)^3 =  1 + 6*x + 27*x^2 + 112*x^3 + 480*x^4 + 2484*x^5 +...

%e where A(x) = 1 + x*[d/dx x*A(x)^3]/A(x)^4.

%e 1/A(x) = 1 - x - x^2 - 2*x^3 - 7*x^4 - 45*x^5 - 468*x^6 - 6624*x^7 -...

%e A'(x)/A(x)^2 = 1 + 2*x + 6*x^2 + 28*x^3 + 225*x^4 + 2808*x^5 +...

%e where A(x) = 1 + x/A(x) + 3*x^2*A'(x)/A(x)^2.

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*deriv(x*A^3)/(A+x*O(x^n))^4);polcoeff(A,n)}

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

%Y Cf. A112936, A112938.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 04 2013