A214765 G.f. satisfies: A(x) = 1/A(-x*A(x)^5).

1, 2, 12, 84, 616, 4832, 42112, 410368, 4316800, 46899648, 512004480, 5554843904, 59657443584, 633013100288, 6639969848320, 69332566233088, 733169635126272, 8068863012833280, 95049764691595264, 1213724245095528448, 16619899465108049920, 238054738089559379968

Offset: 0

Author

Paul D. Hanna, Jul 29 2012

Keywords

• nonn

Comments

Compare g.f. to: G(x) = 1/G(-x*G(x)^5) when G(x) = 1 + x*G(x)^3 (A001764).

An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^5); for example, (*) is satisfied by G(x) = F(m*x) = 1 + m*x*F(m*x)^3 for all m, where F(x) is the g.f. of A001764.

Examples

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 84*x^3 + 616*x^4 + 4832*x^5 + 42112*x^6 +...
A(x)^3 = 1 + 6*x + 48*x^2 + 404*x^3 + 3432*x^4 + 29808*x^5 + 271056*x^6 +...
A(x)^5 = 1 + 10*x + 100*x^2 + 980*x^3 + 9400*x^4 + 89632*x^5 + 866080*x^6 +...
		

Crossrefs

Cf. A214761, A214762, A214763, A214764, A214766, A214767, A214768, A214769.

Programs

PARI

{a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^5+x*O(x^n)))/2);polcoeff(A,n)}
for(n=0,31,print1(a(n),", "))

Formula

The g.f. of this sequence is the limit of the recurrence:

(*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^5))/2 starting at G_0(x) = 1+2*x.