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%I #37 Nov 23 2016 16:45:39
%S 1,2,9,100,1205,18006,350077,8088536,211371561,6176234890,
%T 200898827921,7219180413732,284177412817597,12162803253287246,
%U 562046000617917285,27867599169654763696,1475047571057004959057,83000104748219010488850,4947512767013757600177049,311464596400042198210554620,20652342444419128752639269541,1438800618216725748602640496342
%N 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,...
%C The compositional transpose of functions x*exp(x^n) yields the e.g.f. of A277181.
%H Vaclav Kotesovec, <a href="/A277180/b277180.txt">Table of n, a(n) for n = 1..300</a>
%F 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.
%e 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! +...
%e such that A(x) is the limit of composition of functions x*exp(x^n):
%e 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)
%e working from right to left.
%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( F_1(x)^2 ),
%e F_3(x) = F_2(x) * exp( F_2(x)^3 ),
%e F_4(x) = F_3(x) * exp( F_3(x)^4 ),
%e ...
%e F_{n+1}(x) = F_{n}(x) * exp( F_{n}(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! + 76*x^4/4! + 605*x^5/5! + 5046*x^6/6! +...
%e F_3(x) = x + 2*x^2/2! + 9*x^3/3! + 100*x^4/4! + 1085*x^5/5! + 13686*x^6/6! +...
%e F_4(x) = x + 2*x^2/2! + 9*x^3/3! + 100*x^4/4! + 1205*x^5/5! + 17286*x^6/6! +...
%e ...
%e RELATED SERIES.
%e The logarithm of A(x)/x begins:
%e 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! +...
%e The series reversion of the e.g.f. begins:
%e 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! +...
%o (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))}
%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 = subst(A,x, x*exp(x^(n-i+1) +x*O(x^n))))); n!*polcoeff(A, n)}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A277182 (log A(x)/x), A277181, A136751.
%Y Cf. A278332.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Oct 04 2016
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