A214654 - OEIS

%I #30 Feb 21 2014 03:30:26

%S 1,1,4,29,307,4288,74511,1550203,37588412,1041217039,32446359005,

%T 1123624632224,42814687805649,1780347364682777,80231627759556196,

%U 3895332991309376213,202713074683790193475,11256955024502873008864,664444403260495390747071

%N E.g.f. A(x) satisfies: A'(x) = exp( A(x)*A'(x) ).

%C Compare to the trivial identity: G'(x) = exp(G(x)) when G(x) = -log(1-x).

%H Vaclav Kotesovec, <a href="/A214654/b214654.txt">Table of n, a(n) for n = 1..240</a>

%F E.g.f. A(x) satisfies:

%F (1) A''(x) = A'(x)^3/(1 - A(x)*A'(x)).

%F (2) A(x) = x + Sum_{n>=2} (n-2)^(n-2)*A(x)^n/n!.

%F (3) A'(x) = Sum_{n>=0} (n+1)^(n-1)*A(x)^n/n!.

%F (4) A(x)*A'(x) = Sum_{n>=1} n^(n-1)*A(x)^n/n!.

%F (5) A(x) = Series_Reversion( Integral -x/LambertW(-x) dx ).

%F Let W(x) = Sum_{n>=2} (n-2)^(n-2)*x^n/n!, then e.g.f. A(x) satisfies:

%F (6) A(x) = Series_Reversion(x - W(x)).

%F (7) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) W(x)^n/n!.

%F (8) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) W(x)^n/(n!*x) ).

%F a(n) ~ 2^(2*n-3) * n^(n-2) / (exp(n-2) * (1+exp(-2))^(n-3/2)). - _Vaclav Kotesovec_, Feb 17 2014

%F A(x) = Series_Reversion( -((x^2*(1 + 2*LambertW(-x))) / (4*LambertW(-x)^2)) + 1/4). - _Vaclav Kotesovec_, Feb 21 2014

%e E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 29*x^4/4! + 307*x^5/5! + 4288*x^6/6! +...

%e By definition log(A'(x)) = A(x)*A'(x), where:

%e (3) A'(x) = 1 + x + 4*x^2/2! + 29*x^3/3! + 307*x^4/4! + 4288*x^5/5! +...

%e (4) A(x)*A'(x) = x + 3*x^2/2! + 19*x^3/3! + 185*x^4/4! + 2437*x^5/5! + 40523*x^6/6! + 814355*x^7/7! + 19196769*x^8/8! + 519397829*x^9/9! +...

%e RELATED SERIES:

%e Let W(x) = x^2/2! + x^3/3! + 4*x^4/4! + 27*x^5/5! + 256*x^6/6! + 3125*x^7/7! +...+ (n-2)^(n-2)*x^n/n! +... then

%e (6) A(x) = x + W(A(x)); equivalently, A(x - W(x)) = x.

%e (7) A(x) = x + W(x) + d/dx W(x)^2/2! + d^2/dx^2 W(x)^3/3! + d^3/dx^3 W(x)^4/4! +...

%e (8) log(A(x)/x) = W(x)/x + d/dx W(x)^2/(2!*x) + d^2/dx^2 W(x)^3/(3!*x) + d^3/dx^3 W(x)^4/(4!*x) +...

%t Rest[CoefficientList[InverseSeries[Series[-((x^2*(1 + 2*LambertW[-x])) / (4*LambertW[-x]^2)) + 1/4,{x,0,20}],x],x]*Range[0,20]!] (* _Vaclav Kotesovec_, Feb 21 2014 *)

%o (PARI) {a(n)=local(A=x);for(i=1,n,A=intformal(exp(A*A'+x*O(x^n))));n!*polcoeff(A,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,W=sum(m=2,n+1,(m-2)^(m-2)*x^m/m!)+x*O(x^n)); A=x+sum(m=1, n, Dx(m-1, W^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,W=sum(m=2,n+1,(m-2)^(m-2)*x^m/m!)+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, W^m/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}

%Y Cf. A210949, A214645.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jul 24 2012

%E Program in Mathematica improved by _Vaclav Kotesovec_, Feb 21 2014