A214763 G.f. satisfies: A(x) = 1/A(-x*A(x)^3).

1, 2, 8, 40, 224, 1280, 7168, 40000, 231296, 1436928, 9773056, 72242176, 563679232, 4491707904, 35735001088, 280941652992, 2178641254400, 16710771339264, 127402021142528, 970887186407424, 7436390169329664, 57531833133899776, 451525691751628800, 3608174274928951296

Offset: 0

Author

Paul D. Hanna, Jul 27 2012

Keywords

• nonn

Comments

Compare to: G(x) = 1/G(-x*G(x)^3) when G(x) = 1 + x*G(x)^2 (A000108).

Compare to: B(x) = 1/B(-x*B(x)^3) when B(x) = 1/(1-9*x)^(1/3) = g.f. of A004987.

An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^3); for example, (*) is satisfied by G(x) = C(m*x) = (1-sqrt(1-4*m*x))/(2*m*x) for all m, where C(x) is the Catalan function.

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 40*x^3 + 224*x^4 + 1280*x^5 + 7168*x^6 +...
A(x)^2 = 1 + 4*x + 20*x^2 + 112*x^3 + 672*x^4 + 4096*x^5 + 24640*x^6 +...
A(x)^3 = 1 + 6*x + 36*x^2 + 224*x^3 + 1440*x^4 + 9312*x^5 + 59456*x^6 +...
1/A(x) = A(-x*A(x)^3) = 1 - 2*x - 4*x^2 - 16*x^3 - 80*x^4 - 384*x^5 - 1664*x^6 - 7360*x^7 - 40832*x^8 - 304128*x^9 - 2667008*x^10 -...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A214761, A214762, A214764, A214765, 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^3+x*O(x^n)))/2);polcoeff(A,n)}
for(n=0,30,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)^3))/2 starting at G_0(x) = 1+2*x.