A196018 G.f. satisfies: A(x) = 1/(1-x) - 1/(1-x*A(x)) + 1/(1-x*A(x)^2).

1, 1, 2, 6, 23, 98, 440, 2044, 9742, 47384, 234289, 1174214, 5951877, 30459550, 157168265, 816777857, 4271248777, 22459464722, 118678530165, 629867928597, 3356148860975, 17946684482409, 96280344449069, 518058601390577, 2795121781871727, 15118502434518352

Offset: 0

Author

Paul D. Hanna, Sep 26 2011

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + 98*x^5 + 440*x^6 +...
Related series begin:
1/(1-x*A(x)) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 53*x^5 + 211*x^6 +...
1/(1-x*A(x)^2) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 150*x^5 + 650*x^6 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Programs

PARI

{a(n)=local(A=1+x,X=x+x*O(x^n));for(i=1,n,A=1/(1-X)-1/(1-X*A)+1/(1-X*A^2));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 - x*(A(x) - A(x)^2 - A(x)^3) + x^2*(A(x) - 2*A(x)^2 - A(x)^4) + x^3*A(x)^4.

a(n) ~ sqrt((1/(-1 + r)^2 - s/(-1 + r*s)^2 + s^2/(-1 + r*s^2)^2) / (Pi*(1/(-1 + r*s^2)^2 + r*(1/(-1 + r*s)^3 - (4*s^2) / (-1 + r*s^2)^3)))) / (2*n^(3/2)*r^n), where r = 0.1741099691155951761402154753241071226265020289369... and s = 1.469614426933947254586622522985062658500679266649... are roots of the system of equations 1/(1-r) + 1/(-1+r*s) + 1/(1-r*s^2) = s, 2*r*s / (-1+r*s^2)^2 = 1 + r/(-1+r*s)^2. - Vaclav Kotesovec, Nov 18 2017