A213644 E.g.f. satisfies: A(x) = 1 + x*A(x)^2*exp(x*A(x)).

1, 1, 6, 63, 988, 20725, 546246, 17364445, 646910328, 27652214313, 1334291800330, 71749167806041, 4255000637001588, 275904038948566093, 19420072633921942542, 1474700254793433800805, 120174737260376219862256, 10461031446553525766071249, 968785652772129485926955538

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

Cf. A213643, A006153, A161633.

Sequence in context: A098342 A107995 A166893 * A280476 A295241 A271607

Adjacent sequences: A213641 A213642 A213643 * A213645 A213646 A213647

Keyword

nonn

Author

Paul D. Hanna, Jun 17 2012

Status

approved