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

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A216136 E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^3). 5

%I

%S 1,1,6,75,1456,38560,1296786,52935680,2542934024,140557741488,

%T 8787984793440,613224873661752,47245653830341176,3983499665690137944,

%U 364844394810538703256,36070922050704987248280,3828821598701561543783616,434302348322255060713797120

%N E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^3).

%C More generally, if G(x) = 1/(1 - x*G(x)^p)^(G(x)^q), then

%C [x^n/n! ] G(x)^m = Sum_{k=0..n} m*(p*n+q*k+m)^(k-1) * (-1)^(n-k)*Stirling1(n,k), and

%C [x^n/n! ] log(G(x)) = Sum_{k=1..n} (p*n+q*k)^(k-1) * (-1)^(n-k)*Stirling1(n,k).

%C Generally, for A(x)^A(x) = 1/(1-x*A(x)^p) is limit n->infinity a(n)^(1/n)/n = exp(p*(1-r)/(r-p))*(p-r+exp(r/(p-r))), where r is the root of the equation exp(r/(p-r)) = (r-p)/r*(r + LambertW(-1,-r*exp(-r)). - _Vaclav Kotesovec_, Sep 17 2013

%C Generally, if e.g.f. A(x) satisfies A(x)^A(x) = 1/(1-x*A(x)^p), then a(n) ~ s*sqrt((s^s-1)/(p*(s^s-1)*(p*s^s-1)-s)) * n^(n-1) * (s^(p+s)/(s^s-1))^n / exp(n), where s is the root of the equation (1+log(s))*s = (s^s-1)*p. Compared with my previous result, limit n->infinity a(n)^(1/n)/n = s^(p+s)/(s^s-1)/exp(1). - _Vaclav Kotesovec_, Dec 28 2013

%H Seiichi Manyama, <a href="/A216136/b216136.txt">Table of n, a(n) for n = 0..338</a>

%F (1) a(n) = Sum_{k=0..n} (3*n-k+1)^(k-1)* (-1)^(n-k)* Stirling1(n,k).

%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then

%F (2) a(n,m) = Sum_{k=0..n} m*(3*n-k+m)^(k-1) * (-1)^(n-k) *Stirling1(n,k) ;

%F which is equivalent to the following:

%F (3) a(n,m) = Sum_{k=0..n} m*(3*n-k+m)^(k-1) * {[x^(n-k)] Product_{j=1..n-1} (1+j*x)};

%F (4) a(n,m) = n!*Sum_{k=0..n} m*(3*n-k+m)^(k-1) * {[x^(n-k)] (-log(1-x)/x)^k/k!}.

%F Limit n->infinity a(n)^(1/n)/n = exp(3*(1-r)/(r-3))*(3-r+exp(r/(3-r))) = 2.685525290558..., where r = 0.77397865498224... is the root of the equation exp(r/(3-r)) = (r-3)/r*(r + LambertW(-1,-r*exp(-r)). - _Vaclav Kotesovec_, Sep 17 2013

%F a(n) ~ s*sqrt((s^s-1)/(3*(s^s-1)*(3*s^s-1)-s)) * n^(n-1) * (s^(3+s)/(s^s-1))^n / exp(n), where s = 1.4158017407588097722625060603... is the root of the equation (1+log(s))*s = 3*(s^s-1). - _Vaclav Kotesovec_, Dec 28 2013

%e E.g.f. A(x) = 1 + x + 6*x^2/2! + 75*x^3/3! + 1456*x^4/4! + 38560*x^5/5! +...

%e where

%e A(x)^A(x) = 1 + x + 8*x^2/2! + 114*x^3/3! + 2388*x^4/4! + 66480*x^5/5! +...

%e 1/(1-x*A(x)^3) = 1 + x + 8*x^2/2! + 114*x^3/3! + 2388*x^4/4! + 66480*x^5/5! +...

%t Table[Sum[(3*n-k+1)^(k-1)*(-1)^(n-k)*StirlingS1[n,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Sep 17 2013 *)

%o (PARI) a(n, m=1)=sum(k=0, n, m*(3*n-k+m)^(k-1)*(-1)^(n-k)*stirling(n, k, 1));

%o for(n=0,21,print1(a(n),", "))

%o (PARI) {a(n, m=1)=sum(k=0, n, m*(3*n-k+m)^(k-1)*polcoeff(prod(j=1, n-1, 1+j*x), n-k))}

%o for(n=0,21,print1(a(n),", "))

%o (PARI) {a(n)=local(A=1+x); for(i=0, n, A=exp(-log(1-x*(A^3+x*O(x^n)))/A)); n!*polcoeff(A, n)}

%o for(n=0,21,print1(a(n),", "))

%Y Cf. A141209, A216135, A229237.

%K nonn,changed

%O 0,3

%A _Paul D. Hanna_, Sep 01 2012

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.

Last modified December 1 09:18 EST 2021. Contains 349426 sequences. (Running on oeis4.)