login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A213644 E.g.f. satisfies: A(x) = 1 + x*A(x)^2*exp(x*A(x)). 5
1, 1, 6, 63, 988, 20725, 546246, 17364445, 646910328, 27652214313, 1334291800330, 71749167806041, 4255000637001588, 275904038948566093, 19420072633921942542, 1474700254793433800805, 120174737260376219862256, 10461031446553525766071249, 968785652772129485926955538 (list; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 25 08:33 EST 2021. Contains 341606 sequences. (Running on oeis4.)