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A277180 E.g.f.: A(x) = ... x*exp(x^4) o x*exp(x^3) o x*exp(x^2) o x*exp(x), the composition of functions x*exp(x^n) for n = 1,2,3,... 4
1, 2, 9, 100, 1205, 18006, 350077, 8088536, 211371561, 6176234890, 200898827921, 7219180413732, 284177412817597, 12162803253287246, 562046000617917285, 27867599169654763696, 1475047571057004959057, 83000104748219010488850, 4947512767013757600177049, 311464596400042198210554620, 20652342444419128752639269541, 1438800618216725748602640496342 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The compositional transpose of functions x*exp(x^n) yields the e.g.f. of A277181.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..300

FORMULA

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

EXAMPLE

E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 100*x^4/4! + 1205*x^5/5! + 18006*x^6/6! + 350077*x^7/7! + 8088536*x^8/8! + 211371561*x^9/9! + 6176234890*x^10/10! + 200898827921*x^11/11! + 7219180413732*x^12/12! +...

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

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

working from right to left.

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( F_1(x)^2 ),

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

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

...

F_{n+1}(x) = F_{n}(x) * exp( F_{n}(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! + 76*x^4/4! + 605*x^5/5! + 5046*x^6/6! +...

F_3(x) = x + 2*x^2/2! + 9*x^3/3! + 100*x^4/4! + 1085*x^5/5! + 13686*x^6/6! +...

F_4(x) = x + 2*x^2/2! + 9*x^3/3! + 100*x^4/4! + 1205*x^5/5! + 17286*x^6/6! +...

...

RELATED SERIES.

The logarithm of A(x)/x begins:

log(A(x)/x) = x + 2*x^2/2! + 18*x^3/3! + 144*x^4/4! + 1660*x^5/5! + 27480*x^6/6! + 548394*x^7/7! + 12402992*x^8/8! + 316789848*x^9/9! + 9158652720*x^10/10! + 296955697390*x^11/11! + 10666960742328*x^12/12! +...+ A277182(n)*x^n/n! +...

The series reversion of the e.g.f. begins:

Series_Reversion(A(x)) = x - 2*x^2/2! + 3*x^3/3! - 40*x^4/4! + 505*x^5/5! - 4776*x^6/6! + 53179*x^7/7! - 1065296*x^8/8! + 25478289*x^9/9! - 480072880*x^10/10! + 9400182451*x^11/11! - 300620572968*x^12/12! +...

PROG

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

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

CROSSREFS

Cf. A277182 (log A(x)/x), A277181, A136751.

Cf. A278332.

Sequence in context: A027686 A187647 A322645 * A013520 A041239 A098610

Adjacent sequences:  A277177 A277178 A277179 * A277181 A277182 A277183

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 04 2016

STATUS

approved

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Last modified August 12 23:19 EDT 2020. Contains 336440 sequences. (Running on oeis4.)