A277181 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.

1, 2, 9, 76, 605, 7326, 97237, 1414904, 24130521, 467773210, 9636459041, 215484787332, 5351427245749, 141098897750006, 3995090542811565, 120415709525270896, 3833710980240095537, 130061101059127375794, 4649348119132468282681, 174231442774945244111420, 6859230825811289134828941, 282654139723294546295799502, 12162998707984268597918477189, 546138551651775603897277518696

Offset

1,2

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

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,...

Example

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! +...
such that A(x) is the limit of composition of functions x*exp(x^n):
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 ...
working from left to right.
Illustration of generating method.
Start with F_0(x) = x and then continue as follows.
F_1(x) = x*exp(x),
F_2(x) = F_1( x*exp(x^2) ),
F_3(x) = F_2( x*exp(x^3) ),
F_4(x) = F_3( x*exp(x^4) ),
F_5(x) = F_4( x*exp(x^5) ),
...
F_{n+1}(x) = F_{n}( x*exp(x^(n+1)) ),
...
the limit of which equals the e.g.f. A(x).
The above series begin:
F_1(x) = x + 2*x^2/2! + 3*x^3/3! + 4*x^4/4! + 5*x^5/5! + 6*x^6/6! +...
F_2(x) = x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 245*x^5/5! + 1926*x^6/6! +...
F_3(x) = x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 485*x^5/5! + 5166*x^6/6! +...
F_4(x) = x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 605*x^5/5! + 6606*x^6/6! +...
F_5(x) = x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 605*x^5/5! + 7326*x^6/6! +...
...
A related series begins:
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! +...

Prog

(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)}
for(n=1, 30, print1(a(n), ", "))
(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)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

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

Cf. A278332.

Sequence in context: A335378 A232471 A357539 * A105785 A245406 A337558

Adjacent sequences: A277178 A277179 A277180 * A277182 A277183 A277184

Keyword

nonn

Author

Paul D. Hanna, Oct 04 2016

Status

approved