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%I #16 Nov 22 2021 02:38:16
%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
%O 0,3
%A _Paul D. Hanna_, Sep 01 2012
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