OFFSET
0,3
COMMENTS
Generally, if g.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^p]/A(x)^p, then a(n) ~ (n-1)! * p^(n-1) / (Gamma((p-1)/p) * n^(1/p)). - Vaclav Kotesovec, Feb 22 2014
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
G.f. satisfies: A(x) = 1+x + 6*x^2*[d/dx A(x)]/A(x) (log derivative).
G.f.: A(x) = 1+x+6*x^2/(1-11*x-6*2*5*x^2/(1-23*x-6*3*11*x^2/(1-35*x -6*4*17*x^2/(1-47*x- ... -6*n*(6*n-7)*x^2/(1-(12*n-1)*x - ...)))) (continued fraction).
G.f.: A(x) = 1/(1-1*x/(1-5*x/(1-6*x/(1-11*x/(1-12*x/(1-17*x/(1-18*x/(1 -...)))))))) (continued fraction).
G.f.: G(0) where G(k) = 1 - x*(6*k-1)/( 1 - 6*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 24 2013
a(n) ~ (n-1)! * 6^(n-1) / (Gamma(5/6) * n^(1/6)). - Vaclav Kotesovec, Feb 22 2014
EXAMPLE
A(x) = 1 + x + 6*x^2 + 66*x^3 + 1086*x^4 + 24186*x^5 +...
1/A(x) = 1 - x - 5*x^2 - 55*x^3 - 935*x^4 -... -A008543(n)*x^(n+1)-...
MATHEMATICA
CoefficientList[Series[1/(1 + 1/6*ExpIntegralE[5/6, -1/(6*x)]/E^(1/(6*x))), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 22 2014 *)
PROG
(PARI) {a(n)=local(F=1+x+x*O(x^n)); for(i=1, n, F=1+x+6*x^2*deriv(F)/F); return(polcoeff(F, n, x))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 09 2005
STATUS
approved