login
E.g.f.: A(x) = x*exp(x) o x*exp(x^2) o x*exp(x^3) o x*exp(x^4) o ..., the composition of functions x*exp(x^n) for n=...,3,2,1.
4

%I #25 Nov 23 2016 16:46:02

%S 1,2,9,76,605,7326,97237,1414904,24130521,467773210,9636459041,

%T 215484787332,5351427245749,141098897750006,3995090542811565,

%U 120415709525270896,3833710980240095537,130061101059127375794,4649348119132468282681,174231442774945244111420,6859230825811289134828941,282654139723294546295799502,12162998707984268597918477189,546138551651775603897277518696

%N E.g.f.: A(x) = x*exp(x) o x*exp(x^2) o x*exp(x^3) o x*exp(x^4) o ..., the composition of functions x*exp(x^n) for n=...,3,2,1.

%C The compositional transpose of functions x*exp(x^n) yields the e.g.f. of A277180.

%H Paul D. Hanna, <a href="/A277181/b277181.txt">Table of n, a(n) for n = 1..300</a>

%F E.g.f. A(x) satisfies: Series_Reversion(A(x)) = ... (LambertW(4*x^4)/4)^(1/4) o (LambertW(3*x^3)/3)^(1/3) o (LambertW(2*x^2)/2)^(1/2) o LambertW(x), the composition of functions (LambertW(n*x^n)/n)^(1/n) for n = 1,2,3,...

%e E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 605*x^5/5! + 7326*x^6/6! + 97237*x^7/7! + 1414904*x^8/8! + 24130521*x^9/9! + 467773210*x^10/10! + 9636459041*x^11/11! + 215484787332*x^12/12! +...

%e such that A(x) is the limit of composition of functions x*exp(x^n):

%e A(x) = x*exp(x) o x*exp(x^2) o x*exp(x^3) o x*exp(x^4) o x*exp(x^5) o ...

%e working from left to right.

%e Illustration of generating method.

%e Start with F_0(x) = x and then continue as follows.

%e F_1(x) = x*exp(x),

%e F_2(x) = F_1( x*exp(x^2) ),

%e F_3(x) = F_2( x*exp(x^3) ),

%e F_4(x) = F_3( x*exp(x^4) ),

%e F_5(x) = F_4( x*exp(x^5) ),

%e ...

%e F_{n+1}(x) = F_{n}( x*exp(x^(n+1)) ),

%e ...

%e the limit of which equals the e.g.f. A(x).

%e The above series begin:

%e F_1(x) = x + 2*x^2/2! + 3*x^3/3! + 4*x^4/4! + 5*x^5/5! + 6*x^6/6! +...

%e F_2(x) = x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 245*x^5/5! + 1926*x^6/6! +...

%e F_3(x) = x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 485*x^5/5! + 5166*x^6/6! +...

%e F_4(x) = x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 605*x^5/5! + 6606*x^6/6! +...

%e F_5(x) = x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 605*x^5/5! + 7326*x^6/6! +...

%e ...

%e A related series begins:

%e Series_Reversion(A(x)) = x - 2*x^2/2! + 3*x^3/3! - 16*x^4/4! + 385*x^5/5! - 6696*x^6/6! + 104419*x^7/7! - 1785344*x^8/8! + 37367649*x^9/9! - 986989600*x^10/10! + 30811625251*x^11/11! - 1031073660288*x^12/12! +...

%o (PARI) {a(n) = my(A=x +x*O(x^n)); if(n<=0, 0, for(i=1, n, A = subst(A,x, x*exp(x^i +x*O(x^n))))); n!*polcoeff(A, n)}

%o for(n=1,30,print1(a(n),", "))

%o (PARI) {a(n) = my(A=x+x*O(x^n)); if(n<=0, 0, for(i=1, n, A = A*exp(A^(n-i+1)))); n!*polcoeff(A, n)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A277183 (log(A(x)/x)), A277180, A136751.

%Y Cf. A278332.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 04 2016