A201465 E.g.f. satisfies: A(x) = (x + 2*exp(A(x)) - 2)/3.

1, 2, 14, 162, 2622, 54546, 1386702, 41660226, 1444071006, 56728401138, 2490626473326, 120858220146978, 6423145784929278, 371046277074303954, 23148851187463826958, 1551182540888019542274, 111111330526583477368734, 8472364399282482984295602, 685178683361064789536947374

Offset

1,2

Links

Table of n, a(n) for n=1..19.

Formula

E.g.f. A(x) satisfies: x = A( 2 + 3*x - 2*exp(x) ).

a(n)=(sum(k=0..n-1, (n+k-1)!*sum(j=0..k, 1/(k-j)!*sum(i=0..j, (3^i*(-1)^(i)*2^(j-i)*stirling2(n+j-i-1,j-i))/(i!*(n+j-i-1)!))))), n>0. [From Vladimir Kruchinin, Feb 04 2012]

exp(A(x))-1 is the compositional inverse of 3*log(1+x)-2*x and is the e.g.f. of A058562. - Peter Bala, Jul 12 2012

G.f.: 1/Q(0), where Q(k)= 1 - k*x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013

E.g.f.: (x-2)/3 - LambertW(-2/3*exp((x-2)/3)). - Vaclav Kotesovec, Dec 26 2013

a(n) ~ n^(n-1) / (sqrt(3) * exp(n) * (3*log(3)-3*log(2)-1)^(n-1/2)). - Vaclav Kotesovec, Dec 26 2013

O.g.f.: Sum_{n>=0} 2^n / Product_{k=0..n} (3 - k*x). - Paul D. Hanna, Oct 27 2014

Example

E.g.f.: A(x) = x + 2*x^2 + 14*x^3/3! + 162*x^4/4! + 2622*x^5/5! + 54546*x^6/6! +...
The exponential of the e.g.f. begins:
exp(A(x)) = 1 + x + 3*x^2/2! + 21*x^3/3! + 243*x^4/4! + 3933*x^5/5! + 81819*x^6/6! +...
where x = 2 + 3*A(x) - 2*exp(A(x)).
...
O.g.f.: G(x) = 1 + 2*x + 14*x^2 + 162*x^3 + 2622*x^4 + 54546*x^5 +...
where
G(x) = 1/3 + 2/(3*(3-x)) + 2^2/(3*(3-x)*(3-2*x)) + 2^3/(3*(3-x)*(3-2*x)*(3-3*x)) + 2^4/(3*(3-x)*(3-2*x)*(3-3*x)*(3-4*x)) + 2^5/(3*(3-x)*(3-2*x)*(3-3*x)*(3-4*x)*(3-5*x)) +...

Mathematica

Rest[CoefficientList[1 + InverseSeries[Series[2 + 3*x - 2*Exp[x], {x, 0, 20}], x], x]* Range[0, 20]!] (* Vaclav Kotesovec, Dec 26 2013 *)

Prog

(PARI) {a(n)=n!*polcoeff(serreverse(2+3*x - 2*exp(x+x^2*O(x^n))), n)}
for(n=0, 25, print1(round(A[n+1]), ", "))
(PARI) \p100 \\ set precision
{A=Vec(sum(n=0, 600, 1.*2^n/prod(k=0, n, 3 - k*x + O(x^31))))}
for(n=0, 25, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 27 2014
(Maxima) a(n):=(sum((n+k-1)!*sum(1/(k-j)!*sum((3^i*(-1)^(i)*2^(j-i)*stirling2(n+j-i-1, j-i))/(i!*(n+j-i-1)!), i, 0, j), j, 0, k), k, 0, n-1)); [From Vladimir Kruchinin, Feb 04 2012]

Crossrefs

Cf. variants: A000311, A201466. A058562.

Sequence in context: A354241 A364398 A074635 * A245894 A277362 A209937

Adjacent sequences: A201462 A201463 A201464 * A201466 A201467 A201468

Keyword

nonn

Author

Paul D. Hanna, Dec 01 2011

Status

approved