|
%I #17 Nov 22 2021 02:37:42
%S 1,1,4,33,424,7440,165846,4487966,142930376,5237697744,217106129040,
%T 10043789510832,513016686849624,28676264198255856,1741205465305623240,
%U 114124985340571809480,8030944551164700156096,603905270121593669417472,48328182913534662635924544
%N E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^2).
%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, 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. - _Vaclav Kotesovec_, Dec 28 2013
%H Seiichi Manyama, <a href="/A216135/b216135.txt">Table of n, a(n) for n = 0..357</a>
%F (1) a(n) = Sum_{k=0..n} (2*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*(2*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*(2*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*(2*n-k+m)^(k-1) * {[x^(n-k)] (-log(1-x)/x)^k/k!}.
%F Limit n->infinity a(n)^(1/n)/n = exp(2*(1-r)/(r-2))*(2-r+exp(r/(2-r))) = 1.7802115440907..., where r = 0.655269699533064... is the root of the equation exp(r/(2-r)) = (r-2)/r*(r + LambertW(-1,-r*exp(-r)). - _Vaclav Kotesovec_, Sep 17 2013
%F a(n) ~ s*sqrt((s^s-1)/(2*(s^s-1)*(2*s^s-1)-s)) * n^(n-1) * (s^(2+s)/(s^s-1))^n / exp(n), where s = 1.627893875694537903318580987... is the root of the equation (1+log(s))*s = 2*(s^s-1). - _Vaclav Kotesovec_, Dec 28 2013
%e E.g.f. A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 424*x^4/4! + 7440*x^5/5! +...
%e where
%e A(x)^A(x) = 1 + x + 6*x^2/2! + 60*x^3/3! + 864*x^4/4! + 16360*x^5/5! +...
%e 1/(1-x*A(x)^2) = 1 + x + 6*x^2/2! + 60*x^3/3! + 864*x^4/4! + 16360*x^5/5! +...
%t Table[Sum[(2*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*(2*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*(2*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^2+x*O(x^n)))/A)); n!*polcoeff(A, n)}
%o for(n=0,21,print1(a(n),", "))
%Y Cf. A141209, A216136, A229237.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 01 2012
| 