A232690 E.g.f. satisfies: A(x) = exp( 1/A(x) * Integral A(x)^3 dx ).

1, 1, 2, 7, 33, 202, 1495, 13107, 132062, 1508629, 19227687, 270818542, 4173948097, 69906444393, 1263811926338, 24534217063999, 508951297964193, 11236656534791578, 263054502440239639, 6508910392250017611, 169727899004807970782, 4652123984505282141277, 133711980572082349859559

Offset: 0

Author

Paul D. Hanna, Dec 06 2013

Keywords

• nonn

Comments

Note that G(x) = exp(1/G(x) * Integral G(x)^2 dx) has negative coefficients.

Compare e.g.f. to: B(x) = exp( 1/B(x) * Integral B(x) dx ) where B(y) = Bessel polynomial y_n(-1) (cf. A000806).

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 33*x^4/4! + 202*x^5/5! +...
Related expansions:
log(A(x)) = x + x^2/2! + 3*x^3/3! + 11*x^4/4! + 61*x^5/5! + 393*x^6/6! +...
Integral A(x)^3 dx = x + 3*x^2/2! + 12*x^3/3! + 63*x^4/4! + 411*x^5/5! +...
1/A(x) = 1 - x - x^3/3! - x^4/4! - 12*x^5/5! - 41*x^6/6! - 451*x^7/7! -...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A232691, A232692.

Programs

Maple

seq(n! * coeff(series(sqrt(LambertW(-1,(4*x-3)*exp(-3))/(4*x-3)), x, n+1), x, n), n=0..20); # Vaclav Kotesovec, Jan 05 2014

Mathematica

CoefficientList[FullSimplify[Assuming[Element[x, Reals], Series[Sqrt[LambertW[-1,(4*x-3)*E^(-3)]/(4*x-3)], {x, 0, 20}]]], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 05 2014 *)

PARI

{a(n)=local(A=1+x);for(i=1,n,A=exp(1/A*intformal(A^3+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f.: sqrt(LambertW(-1,(4*x-3)*exp(-3))/(4*x-3)). - Vaclav Kotesovec, Jan 05 2014

Limit n->infinity (a(n)/n!)^(1/n) = 4/3. - Vaclav Kotesovec, Jan 05 2014