A218168 G.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^3]/A(x)^4.

1, 1, 2, 5, 16, 77, 630, 7956, 132480, 2664389, 62140078, 1643056625, 48503833280, 1580192424580, 56292381309608, 2176251789724500, 90726545080501440, 4056755120695005717, 193649641793538259950, 9828338731644277922679, 528459404673048692682384

Offset: 0

Author

Paul D. Hanna, Mar 04 2013

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 77*x^5 + 630*x^6 + 7956*x^7 +...
Related expansions:
A'(x) = 1 + 4*x + 15*x^2 + 64*x^3 + 385*x^4 + 3780*x^5 + 55692*x^6 +...
A(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 46*x^4 + 206*x^5 + 1503*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 96*x^4 + 414*x^5 + 2735*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 48*x^3 + 173*x^4 + 736*x^5 + 4486*x^6 +...
d/dx x*A(x)^3 =  1 + 6*x + 27*x^2 + 112*x^3 + 480*x^4 + 2484*x^5 +...
where A(x) = 1 + x*[d/dx x*A(x)^3]/A(x)^4.
1/A(x) = 1 - x - x^2 - 2*x^3 - 7*x^4 - 45*x^5 - 468*x^6 - 6624*x^7 -...
A'(x)/A(x)^2 = 1 + 2*x + 6*x^2 + 28*x^3 + 225*x^4 + 2808*x^5 +...
where A(x) = 1 + x/A(x) + 3*x^2*A'(x)/A(x)^2.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Crossrefs

Cf. A112936, A112938.

Programs

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)}
for(n=0,30,print1(a(n),", "))

Formula

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

a(n) ~ c * 3^n * n! / n^2, where c = 0.0220701841954367321830819129913... - Vaclav Kotesovec, Aug 24 2017