A214654 E.g.f. A(x) satisfies: A'(x) = exp( A(x)*A'(x) ).

1, 1, 4, 29, 307, 4288, 74511, 1550203, 37588412, 1041217039, 32446359005, 1123624632224, 42814687805649, 1780347364682777, 80231627759556196, 3895332991309376213, 202713074683790193475, 11256955024502873008864, 664444403260495390747071

Offset

1,3

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..240

Formula

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

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

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

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

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

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

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

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

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

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

a(n) ~ 2^(2*n-3) * n^(n-2) / (exp(n-2) * (1+exp(-2))^(n-3/2)). - Vaclav Kotesovec, Feb 17 2014

A(x) = Series_Reversion( -((x^2*(1 + 2*LambertW(-x))) / (4*LambertW(-x)^2)) + 1/4). - Vaclav Kotesovec, Feb 21 2014

Example

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! +...
By definition log(A'(x)) = A(x)*A'(x), where:
(3) A'(x) = 1 + x + 4*x^2/2! + 29*x^3/3! + 307*x^4/4! + 4288*x^5/5! +...
(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! +...
RELATED SERIES:
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
(6) A(x) = x + W(A(x)); equivalently, A(x - W(x)) = x.
(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! +...
(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) +...

Mathematica

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 *)

Prog

(PARI) {a(n)=local(A=x); for(i=1, n, A=intformal(exp(A*A'+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{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)}
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{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)}

Crossrefs

Cf. A210949, A214645.

Sequence in context: A089470 A360834 A349599 * A357321 A067146 A210949

Adjacent sequences: A214651 A214652 A214653 * A214655 A214656 A214657

Keyword

nonn

Author

Paul D. Hanna, Jul 24 2012

Extensions

Program in Mathematica improved by Vaclav Kotesovec, Feb 21 2014

Status

approved