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%I #9 Nov 18 2016 06:52:48
%S 1,2,12,48,540,5040,53760,745920,12897360,206236800,3682657440,
%T 83366236800,1849952744640,45496897205760,1176291493977600,
%U 32097739145472000,983522184476832000,31186123650255974400,1021800029809932172800,35859334900163943168000,1320567044275182140236800,50849016060752080883712000,2063131407115000462188748800,86946484539570696351914803200
%N E.g.f.: logarithm of G(x)/x where G(x) = x*exp(x) o x*exp(x^2) o x*exp(x^3) o x*exp(x^4) o ..., a composition of functions x*exp(x^n) for n = ...,3,2,1.
%C E.g.f. equals the logarithm of G(x)/x, where G(x) equals the e.g.f. of A277181.
%e E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 48*x^4/4! + 540*x^5/5! + 5040*x^6/6! + 53760*x^7/7! + 745920*x^8/8! + 12897360*x^9/9! + 206236800*x^10/10! + 3682657440*x^11/11! + 83366236800*x^12/12! + 1849952744640*x^13/13! + 45496897205760*x^14/14! + 1176291493977600*x^15/15! + 32097739145472000*x^16/16! +...
%e such that x*exp(A(x)) equals the infinite composition of functions:
%e x*exp(A(x)) = x*exp(x) o x*exp(x^2) o x*exp(x^3) o x*exp(x^4) o ...,
%e which expands to begin:
%e x*exp(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! +...+ A277181(n)*x^n/n! +...
%e A related series expansion begins
%e exp(A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 121*x^4/4! + 1221*x^5/5! + 13891*x^6/6! + 176863*x^7/7! + 2681169*x^8/8! + 46777321*x^9/9! + 876041731*x^10/10! + 17957065611*x^11/11! + 411648249673*x^12/12! +...
%e GENERATING METHOD.
%e One can generate the e.g.f. by the following process.
%e Find A(x) = L_1 by working backward using the relation
%e L_1 = L_2 + x^1*exp(1*L_2)
%e L_2 = L_3 + x^2*exp(2*L_3)
%e L_3 = L_4 + x^3*exp(3*L_4)
%e L_4 = L_5 + x^4*exp(4*L_5)
%e ...
%e L_{n} = L_{n+1} + x^n*exp( n*L_{n+1} )
%e ...
%e Explicitly, the initial series begin:
%e L_1 = x + x^2 + 2*x^3 + 2*x^4 + 9/2*x^5 + 7*x^6 + 32/3*x^7 + 37/2*x^8 +...
%e L_2 = x^2 + x^3 + x^4 + 3*x^5 + 3*x^6 + 6*x^7 + 8*x^8 + 20*x^9 + 22*x^10 +...
%e L_3 = x^3 + x^4 + x^5 + x^6 + 4*x^7 + 4*x^8 + 8*x^9 + 8*x^10 + 35/2*x^11 +...
%e L_4 = x^4 + x^5 + x^6 + x^7 + x^8 + 5*x^9 + 5*x^10 + 10*x^11 + 10*x^12 +...
%e L_5 = x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + 6*x^11 + 6*x^12 + 12*x^13 + 12*x^14 +...
%e L_6 = x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + 7*x^13 + 7*x^14 + 14*x^15 + 14*x^16 +...
%e L_7 = x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + 8*x^15 + 8*x^16 + 16*x^17 + 16*x^18 +...
%e L_8 = x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + 9*x^17 + 9*x^18 + 18*x^19 + 18*x^20 +...
%e ...
%o (PARI) {a(n) = my(A=x +x^2*O(x^n)); if(n<=0, 0, for(i=1, n, A = subst(A, x, x*exp(x^i +x^2*O(x^n))))); n!*polcoeff(log(A/x), n)}
%o for(n=1, 30, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=x +x^2*O(x^n)); if(n<=0, 0, for(i=1, n, A = A*exp(A^(n-i+1)))); n!*polcoeff(log(A/x), n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A277181 (x*exp(A(x)), A277182.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Oct 07 2016
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