A198887 E.g.f. satisfies: A(x) = exp(x*A(x)^2*A(-x)).

1, 1, 3, 28, 269, 5056, 84247, 2400448, 57253849, 2191568896, 68151324491, 3278448139264, 125802549088933, 7291045162516480, 332950230966532831, 22581201334925049856, 1196122595530554458033, 92934371464549349982208, 5602230959364892208231443

Offset: 0

Author

Paul D. Hanna, Oct 30 2011

Keywords

• nonn

Comments

Limit n->infinity (a(n)/n!)^(1/n) = 1/r = 4.263493124332896881178517333221505445574016761952741537891924..., where r = 0.234549457648408586261093274213550311973... and s = 1.724680091765540585933497362883851976875... are roots of the system of equations s*sqrt((r*s*LambertW(2*r*s))/2) = log(s), s*sqrt((r*s*LambertW(2*r*s))/2)*(4 + 3*LambertW(2*r*s)) = 2*(1 + LambertW(2*r*s)). - Vaclav Kotesovec, Jul 16 2014

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 28*x^3/3! + 269*x^4/4! + 5056*x^5/5! +...
Related series:
A(x)^2*A(-x) = 1 + x + 7*x^2/2! + 40*x^3/3! + 709*x^4/4! + 8016*x^5/5! +...
log(A(x)) = x + 2*x^2/2! + 21*x^3/3! + 160*x^4/4! + 3545*x^5/5! + 48096*x^6/6! +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..240

Crossrefs

Cf. A143600.

Programs

PARI

{a(n)=local(A=1+x*O(x^n));for(n=0,n,A=exp(x*A^2*subst(A,x,-x)+x*O(x^n)));n!*polcoeff(A,n)}

Formula

E.g.f. satisfies x*y^2*sqrt(LambertW(2*x*y)/(2*x*y)) = log(y), where y = A(x). - Vaclav Kotesovec, Jul 15 2014