cp's OEIS Frontend

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.

%I A277181 #25 Nov 23 2016 16:46:02
%S A277181 1,2,9,76,605,7326,97237,1414904,24130521,467773210,9636459041,
%T A277181 215484787332,5351427245749,141098897750006,3995090542811565,
%U A277181 120415709525270896,3833710980240095537,130061101059127375794,4649348119132468282681,174231442774945244111420,6859230825811289134828941,282654139723294546295799502,12162998707984268597918477189,546138551651775603897277518696
%N 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.
%C A277181 The compositional transpose of functions x*exp(x^n) yields the e.g.f. of A277180.
%H A277181 Paul D. Hanna, <a href="/A277181/b277181.txt">Table of n, a(n) for n = 1..300</a>
%F A277181 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 A277181 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 A277181 such that A(x) is the limit of composition of functions x*exp(x^n):
%e A277181 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 A277181 working from left to right.
%e A277181 Illustration of generating method.
%e A277181 Start with F_0(x) = x and then continue as follows.
%e A277181 F_1(x) = x*exp(x),
%e A277181 F_2(x) = F_1( x*exp(x^2) ),
%e A277181 F_3(x) = F_2( x*exp(x^3) ),
%e A277181 F_4(x) = F_3( x*exp(x^4) ),
%e A277181 F_5(x) = F_4( x*exp(x^5) ),
%e A277181 ...
%e A277181 F_{n+1}(x) = F_{n}( x*exp(x^(n+1)) ),
%e A277181 ...
%e A277181 the limit of which equals the e.g.f. A(x).
%e A277181 The above series begin:
%e A277181 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 A277181 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 A277181 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 A277181 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 A277181 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 A277181 ...
%e A277181 A related series begins:
%e A277181 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 A277181 (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 A277181 for(n=1,30,print1(a(n),", "))
%o A277181 (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 A277181 for(n=1,30,print1(a(n),", "))
%Y A277181 Cf. A277183 (log(A(x)/x)), A277180, A136751.
%Y A277181 Cf. A278332.
%K A277181 nonn
%O A277181 1,2
%A A277181 _Paul D. Hanna_, Oct 04 2016