A232691 E.g.f. satisfies: A(x) = exp( 1/A(x)^2 * Integral A(x)^6 dx ).

1, 1, 3, 19, 161, 1857, 25843, 433891, 8378913, 185022049, 4565674115, 125075024211, 3755498096257, 122872235056993, 4345683577199283, 165338206044981091, 6730088764152273857, 291935651271257092161, 13440846879808207921027, 654704450541594973156627

Offset

0,3

Comments

Note that G(x) = exp(1/G(x)^2 * Integral G(x)^5 dx) has negative coefficients.

Compare e.g.f. to: B(x) = exp( 1/B(x)^2 * Integral B(x)^2 dx ) where B(y) = Bessel polynomial y_n(-2) (cf. A002119).

Links

Table of n, a(n) for n=0..19.

Formula

E.g.f.: (LambertW(-1, (8*x-3)*exp(-3))/(8*x-3))^(1/4). - Vaclav Kotesovec, Jan 05 2014

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 161*x^4/4! + 1857*x^5/5! +...
Related expansions:
log(A(x)) = x + 2*x^2/2! + 12*x^3/3! + 88*x^4/4! + 976*x^5/5! + 12576*x^6/6! +...
Integral A(x)^6 dx = x + 6*x^2/2! + 48*x^3/3! + 504*x^4/4! + 6576*x^5/5! +...
1/A(x)^2 = 1 - 2*x - 8*x^3/3! - 16*x^4/4! - 384*x^5/5! - 2624*x^6/6! -...

Maple

seq(n! * coeff(series((LambertW(-1, (8*x-3)*exp(-3))/(8*x-3))^(1/4), x, n+1), x, n), n=0..20) # Vaclav Kotesovec, Jan 05 2014

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(1/A^2*intformal(A^6+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A232690, A232692.

Sequence in context: A301921 A054765 A362660 * A057719 A289258 A199559

Adjacent sequences: A232688 A232689 A232690 * A232692 A232693 A232694

Keyword

nonn

Author

Paul D. Hanna, Dec 06 2013

Status

approved