|
%I #23 Dec 28 2013 13:50:01
%S 1,2,18,252,4940,124350,3823722,138915560,5822192952,276522143130,
%T 14677209803630,860990013672492,55315008281020644,3862656545279925302,
%U 291301089508829138130,23595204076694940812880,2042970533426395737658352,188298566037963463789282482
%N E.g.f. satisfies: A(x) = x + A(x)^2*exp(A(x)).
%H Vincenzo Librandi, <a href="/A213643/b213643.txt">Table of n, a(n) for n = 1..200</a>
%F E.g.f.: A(x) = log(G(x)) where G(x) = exp(x*Catalan(x*G(x))) is the e.g.f. of A161629, and Catalan(x) = (1-sqrt(1-4*x))/(2*x).
%F E.g.f.: Series_Reversion(x - x^2*exp(x)).
%F E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) exp(n*x)*x^(2*n) / n!.
%F E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) exp(n*x)*x^(2*n-1) / n! ).
%F O.g.f.: Sum_{n>=0} (2*n)!/n! * x^(n+1) / (1 - n*x)^(2*n+1).
%F a(n) = Sum_{k=0..n-1} k^(n-k-1)/(n-k-1)! * (n+k-1)!/k!.
%F a(n) = n*A213644(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). - _Vaclav Kotesovec_, Jul 13 2013
%F a(n) ~ (1-r) * n^(n-1) * (r*(1+r)/(1-r))^n / (sqrt(r*(1+2*r-r^2))*exp(n)). - _Vaclav Kotesovec_, Dec 28 2013
%e E.g.f.: A(x) = x + 2*x^2/2! + 18*x^3/3! + 252*x^4/4! + 4940*x^5/5! +...
%e where A(x - x^2*exp(x)) = x and A(x) = x + A(x)^2*exp(A(x)).
%e Related expansions:
%e A(x)^2 = 2*x^2/2! + 12*x^3/3! + 168*x^4/4! + 3240*x^5/5! + 80880*x^6/6! +...
%e A(x) = x*Catalan(x*G(x)) where G(x) = exp(A(x)):
%e exp(A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 349*x^4/4! + 6821*x^5/5! + 171421*x^6/6! +..., which is the e.g.f. of A161629.
%e A(x) = x + exp(x)*x^2 + d/dx exp(2*x)*x^4/2! + d^2/dx^2 exp(3*x)*x^6/3! + d^3/dx^3 exp(4*x)*x^8/4! +...
%e log(A(x)/x) = exp(x)*x + d/dx exp(2*x)*x^3/2! + d^2/dx^2 exp(3*x)*x^5/3! + d^3/dx^3 exp(4*x)*x^7/4! +...
%e Ordinary Generating Function:
%e O.g.f.: x + 2*x^2 + 18*x^3 + 252*x^4 + 4940*x^5 + 124350*x^6 +...
%e O.g.f.: x + 2*x^2/(1-x)^3 + 6*2!*x^3/(1-2*x)^5 + 20*3!*x^4/(1-3*x)^7 + 70*4!*x^5/(1-4*x)^9 + 252*5!*x^6/(1-5*x)^11 +...+ (2*n)!/n!*x^(n+1)/(1-n*x)^(2*n+1) +...
%p a:= n-> n!*coeff(series(RootOf(A=x+A^2*exp(A), A), x, n+1), x, n):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jul 18 2013
%t Flatten[{1,Table[Sum[k^(n-k-1)/(n-k-1)!*(n+k-1)!/k!,{k,0,n-1}],{n,2,20}]}] (* _Vaclav Kotesovec_, Jul 13 2013 *)
%o (PARI) {a(n)=sum(k=0,n-1, k^(n-k-1)/(n-k-1)! * (n+k-1)!/k! )}
%o (PARI) {a(n)=n!*polcoeff(serreverse(x-x^2*exp(x+x*O(x^n))),n)}
%o for(n=1,25,print1(a(n),", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, exp(m*x+x*O(x^n))*x^(2*m)/m!)); n!*polcoeff(A, n)}
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, exp(m*x+x*O(x^n))*x^(2*m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
%o (PARI) /* O.g.f.: */
%o {a(n)=polcoeff(sum(m=0,n,(2*m)!/m!*x^(m+1)/(1-m*x+x*O(x^n))^(2*m+1)),n)}
%Y Cf. A213644, A161629, A200319, A215003.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 17 2012
| 