A214762 G.f. satisfies: A(x) = 1/A(-x*A(x)^2).

1, 2, 6, 24, 110, 496, 2156, 9216, 38742, 160032, 664532, 2898848, 13923468, 75361600, 450629592, 2844358656, 18224898790, 116051632704, 728724233988, 4509502911328, 27569637798116, 167072272244352, 1006431412676456, 6037728817690112, 36101656922629500

Offset: 0

Author

Paul D. Hanna, Jul 29 2012

Keywords

• nonn

Comments

Compare to: W(x) = 1/W(-x*W(x)^2) when W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.

Compare to: B(x) = 1/B(-x*B(x)^2) when B(x) = Sum_{n>=0} (2*n)!*x^n/n!^2.

An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^2); for example, (*) is satisfied by G(x) = W(m*x) = LambertW(-m*x)/(-m*x) for all m, where W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 110*x^4 + 496*x^5 + 2156*x^6 +...
Related expansions:
A(x)^2 = 1 + 4*x + 16*x^2 + 72*x^3 + 352*x^4 + 1720*x^5 + 8192*x^6 +...
1/A(x) = A(-x*A(x)^2) = 1 - 2*x - 2*x^2 - 8*x^3 - 34*x^4 - 112*x^5 - 324*x^6 - 896*x^7 - 1866*x^8 - 800*x^9 + 5540*x^10 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..290

Crossrefs

Cf. A214761, A214763, 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^2+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)^2))/2 starting at G_0(x) = 1+2*x.