A213644 - OEIS

%I #25 Oct 27 2024 17:55:35

%S 1,1,6,63,988,20725,546246,17364445,646910328,27652214313,

%T 1334291800330,71749167806041,4255000637001588,275904038948566093,

%U 19420072633921942542,1474700254793433800805,120174737260376219862256,10461031446553525766071249,968785652772129485926955538

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

%H Vincenzo Librandi, <a href="/A213644/b213644.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f. satisfies: A(x) = 1/(1 - x*A(x)*exp(x*A(x))).

%F E.g.f. satisfies: A(x-x^2*exp(x)) = 1/(1-x*exp(x)).

%F a(n) = 1/(n+1) * Sum_{k=0..n} k^(n-k)/(n-k)! * (n+k)!/k!.

%F a(n) = A213643(n+1)/(n+1).

%F 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

%F 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

%e E.g.f.: A(x) = 1 + x + 6*x^2/2! + 63*x^3/3! + 988*x^4/4! + 20725*x^5/5! +...

%e such that A(x-x^2*exp(x)) = 1/(1-x*exp(x)) where:

%e 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! +...

%t 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 *)

%o (PARI) {a(n)=1/(n+1)*sum(k=0, n, k^(n-k)/(n-k)! * (n+k)!/k! )}

%o (PARI) {a(n)=n!*polcoeff((1/x)*serreverse(x-x^2*exp(x+x*O(x^n))),n)}

%o for(n=0,25,print1(a(n),", "))

%Y Cf.  A213643, A006153, A161633.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 17 2012