OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
E.g.f. satisfies: A(x) = 1/(1 - x*A(x)*exp(x*A(x))).
E.g.f. satisfies: A(x-x^2*exp(x)) = 1/(1-x*exp(x)).
a(n) = 1/(n+1) * Sum_{k=0..n} k^(n-k)/(n-k)! * (n+k)!/k!.
a(n) = A213643(n+1)/(n+1).
Limit n->infinity (a(n)/n!)^(1/n) = r*(1+r)/(1-r) = 5.5854662015218413..., where r = 0.7603592340333989... is the root of the equation (1-r^2)/r^2 = exp((r-1)/r), same as for A213643. - Vaclav Kotesovec, Jul 15 2013
a(n) ~ (1+r)*sqrt(r/(1+2*r-r^2)) * n^(n-1) * (r*(1+r)/(1-r))^n / exp(n). - Vaclav Kotesovec, Dec 28 2013
EXAMPLE
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 63*x^3/3! + 988*x^4/4! + 20725*x^5/5! +...
such that A(x-x^2*exp(x)) = 1/(1-x*exp(x)) where:
1/(1-x*exp(x)) = 1 + x + 4*x^2/2! + 21*x^3/3! + 148*x^4/4! + 1305*x^5/5! +...+ A006153(n)*x^n/n! +...
MATHEMATICA
Flatten[{1, Table[1/(n+1)*Sum[k^(n-k)/(n-k)!*(n+k)!/k!, {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 15 2013 *)
PROG
(PARI) {a(n)=1/(n+1)*sum(k=0, n, k^(n-k)/(n-k)! * (n+k)!/k! )}
(PARI) {a(n)=n!*polcoeff((1/x)*serreverse(x-x^2*exp(x+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 17 2012
STATUS
approved