|
%I #18 Nov 14 2017 03:04:37
%S 1,1,2,9,64,620,7626,113792,1997192,40316544,920271840,23438308872,
%T 658947505272,20270099889624,677226678369528,24420959694718680,
%U 945370712175873216,39103903755819561984,1721215383181421110848,80329148928437231089152
%N E.g.f. satisfies: A(x)^A(x) = 1/(1 - x*A(x)).
%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 G. C. Greubel, <a href="/A141209/b141209.txt">Table of n, a(n) for n = 0..385</a>
%F From _Paul D. Hanna_, Jul 08 2009: (Start)
%F (1) a(n) = Sum_{k=0..n} (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*(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*(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*(n-k+m)^(k-1) * {[x^(n-k)] (-log(1-x)/x)^k/k!}.
%F (End)
%F Limit n->infinity a(n)^(1/n)/n = exp((2*r-1)/(1-r))*(1+(1-r)*exp(r/(r-1))) = 0.97848198198076..., where r = 0.42324001455512542... is the root of the equation exp(r/(1-r)) = (r-1)/r*(r + LambertW(-1,-r*exp(-r)). - _Vaclav Kotesovec_, Sep 17 2013
%F a(n) ~ s*sqrt((s^s-1)/((s^s-1)^2-s)) * n^(n-1) * (s^(1+s)/(s^s-1))^n / exp(n), where s = 2.083029805648017585241865819... is the root of the equation (1+log(s))*s = (s^s-1). - _Vaclav Kotesovec_, Dec 28 2013
%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 620*x^5/5! +...
%t Table[Sum[(n-k+1)^(k-1)*Abs[StirlingS1[n,k]], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Sep 17 2013 *)
%t E^((2*r-1)/(1-r))*(1+(1-r)*E^(r/(r-1)))/.FindRoot[E^(r/(1-r))==(r-1)/r*(r+LambertW[-1,-r*E^(-r)]), {r,1/2}, WorkingPrecision->50] (* program for numerical value of the limit n->infinity a(n)^(1/n)/n, _Vaclav Kotesovec_, Sep 17 2013 *)
%o (PARI) {a(n)=local(A=1+x);for(i=0,n,A=exp(-log(1-x*(A+O(x^n)))/A));n!*polcoeff(A,n)}
%o From _Paul D. Hanna_, Jul 08 2009: (Start)
%o (PARI) {a(n,m=1)=sum(k=0,n,m*(n-k+m)^(k-1)*polcoeff(prod(j=1,n-1,1+j*x),n-k))}
%o (PARI) {a(n,m=1)=n!*sum(k=0,n,m*(n-k+m)^(k-1)*polcoeff((-log(1-x+x*O(x^n))/x)^k/k!,n-k))}
%o (PARI) {Stirling1(n,k)=n!*polcoeff(binomial(x,n),k)}
%o {a(n,m=1)=sum(k=0,n,m*(n-k+m)^(k-1)*(-1)^(n-k)*Stirling1(n,k))}
%o (End)
%Y Cf. A216135, A216136, A229237, A008275 (Stirling1), A141209 (A162655), A191908.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 01 2008
|
