A201466 E.g.f. satisfies: A(x) = (x + 3*exp(A(x)) - 3)/4.

1, 3, 30, 498, 11568, 345432, 12606240, 543678672, 27054328512, 1525746223488, 96167433279360, 6699404849841408, 511152613463843328, 42391161255859802112, 3796840836492517125120, 365260399012767192102912, 37561729737177160757133312, 4111876748834828077514170368

Offset

1,2

Links

Table of n, a(n) for n=1..18.

Formula

E.g.f. satisfies: x = A( 3 + 4*x - 3*exp(x) ).

E.g.f.: (x-3)/4 - LambertW(-3*exp((x-3)/4)/4). - Vaclav Kotesovec, Jan 10 2014

a(n) ~ n^(n-1) / (2 * (4*log(4/3)-1)^(n-1/2) * exp(n)). - Vaclav Kotesovec, Jan 10 2014

O.g.f.: Sum_{n>=0} 3^n / Product_{k=0..n} (4 - k*x). - Paul D. Hanna, Oct 27 2014

Example

E.g.f.: A(x) = x + 3*x^2/2! + 30*x^3/3! + 498*x^4/4! + 11568*x^5/5! + 345432*x^6/6! +...
The exponential of the e.g.f. begins:
exp(A(x)) = 1 + x + 4*x^2/2! + 40*x^3/3! + 664*x^4/4! + 15424*x^5/5! + 460576*x^6/6! +...
where x = 3 + 4*A(x) - 3*exp(A(x)).
...
O.g.f.: G(x) = 1 + 3*x + 30*x^2 + 498*x^3 + 11568*x^4 + 345432*x^5 +...
where
G(x) = 1/4 + 3/(4*(4-x)) + 3^2/(4*(4-x)*(4-2*x)) + 3^3/(4*(4-x)*(4-2*x)*(4-3*x)) + 3^4/(4*(4-x)*(4-2*x)*(4-3*x)*(4-4*x)) + 3^5/(4*(4-x)*(4-2*x)*(4-3*x)*(4-4*x)*(4-5*x)) +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[3 - 3*E^x + 4*x, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 10 2014 *)

Prog

(PARI) {a(n)=n!*polcoeff(serreverse(3+4*x - 3*exp(x+x^2*O(x^n))), n)}
(PARI) \p100 \\ set precision
{A=Vec(sum(n=0, 600, 1.*3^n/prod(k=0, n, 4 - k*x + O(x^31))))}
for(n=0, 25, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 27 2014

Crossrefs

Cf. variants: A000311, A201465.

Sequence in context: A164945 A354252 A354261 * A064352 A366002 A338278

Adjacent sequences: A201463 A201464 A201465 * A201467 A201468 A201469

Keyword

nonn

Author

Paul D. Hanna, Dec 01 2011

Status

approved