A214766 G.f. satisfies: A(x) = 1/A(-x*A(x)^6).

1, 2, 14, 112, 910, 8008, 84588, 1059296, 13998070, 179505848, 2193386772, 26007310560, 306461781228, 3616653947520, 42388643986040, 493154764709376, 5905712543971814, 78382075059128216, 1209853310234969668, 20945651586098921696, 378625571347575985092

Offset: 0

Author

Paul D. Hanna, Jul 29 2012

Keywords

• nonn

Comments

Compare to: W(x) = 1/W(-x*W(x)^6) when W(x) = Sum_{n>=0} (3*n+1)^(n-1)*x^n/n!.

An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^6); for example, (*) is satisfied by G(x) = W(m*x), where W(x) = Sum_{n>=0} (3*n+1)^(n-1)*x^n/n!.

Examples

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 112*x^3 + 910*x^4 + 8008*x^5 + 84588*x^6 +...
A(x)^6 = 1 + 12*x + 144*x^2 + 1672*x^3 + 18720*x^4 + 207000*x^5 + 2339072*x^6 +...
		

Crossrefs

Cf. A214761, A214762, A214763, A214764, A214765, A214767, A214768, A214769.

Programs

PARI

{a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^6+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)^6))/2 starting at G_0(x) = 1+2*x.